U.S. patent application number 13/993545 was filed with the patent office on 2014-08-14 for dual composite field advanced encryption standard memory encryption engine.
The applicant listed for this patent is Shay Gueron, Ram K. Krishnamurthy, Sanu K. Mathew. Invention is credited to Shay Gueron, Ram K. Krishnamurthy, Sanu K. Mathew.
Application Number | 20140229741 13/993545 |
Document ID | / |
Family ID | 48698370 |
Filed Date | 2014-08-14 |
United States Patent
Application |
20140229741 |
Kind Code |
A1 |
Mathew; Sanu K. ; et
al. |
August 14, 2014 |
Dual Composite Field Advanced Encryption Standard Memory Encryption
Engine
Abstract
A different set of polynomials may be selected for encryption
and decryption accelerators. That is, different sets of polynomials
are used for encryption and decryption, each set being chosen to
use less area and deliver more power for a memory encryption
engine. This is advantageous in some embodiments since memory read
operations are typically more critical and latency sensitive than
memory writes.
Inventors: |
Mathew; Sanu K.; (Hillsboro,
OR) ; Gueron; Shay; (Haifa, IL) ;
Krishnamurthy; Ram K.; (Portland, OR) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Mathew; Sanu K.
Gueron; Shay
Krishnamurthy; Ram K. |
Hillsboro
Haifa
Portland |
OR
OR |
US
IL
US |
|
|
Family ID: |
48698370 |
Appl. No.: |
13/993545 |
Filed: |
December 30, 2011 |
PCT Filed: |
December 30, 2011 |
PCT NO: |
PCT/US11/68003 |
371 Date: |
April 22, 2014 |
Current U.S.
Class: |
713/189 |
Current CPC
Class: |
H04L 2209/122 20130101;
G06F 21/60 20130101; G06F 21/72 20130101; H04L 9/0631 20130101 |
Class at
Publication: |
713/189 |
International
Class: |
G06F 21/72 20060101
G06F021/72; G06F 21/60 20060101 G06F021/60 |
Claims
1. A method comprising: using a first set of polynomials in a
memory encryption engine for encryption and; using a different set
of polynomials in said engine for decryption.
2. The method of claim 1 including using encryption operations for
reading.
3. The method of claim 1 including using Advanced Encryption
Standard.
4. The method of claim 1 including selecting polynomials to
optimize area usage.
5. The method of claim 1 including selecting polynomials to
optimize power consumption.
6. The method of claim 1 including using Galois polynomials
7. The method of claim 1 including using irreducible
polynomials.
8. The method of claim 1 including locating a primitive element
that is both a generator and a root of a composite field.
9. The method of claim 8 including ensuring that an element exists
in the field such that none of the powers of the element is
one.
10. A non-transitory computer readable medium storing instructions
to enable a processor to: use a first set of polynomials for
encryption and; use a different set of polynomials for
decryption.
11. The medium of claim 10 further storing instructions to use
encryption operations for reading.
12. The medium of claim 10 further storing instructions to use
Advanced Encryption Standard.
13. The medium of claim 10 further storing instructions to select
polynomials to optimize area usage.
14. The medium of claim 10 further storing instructions to select
polynomials to optimize power consumption.
15. The medium of claim 10 further storing instructions to use
Galois polynomials.
16. The medium of claim 10 further storing instructions to use
irreducible polynomials.
17. The medium of claim 10 further storing instructions to locate a
primitive element that is both a generator and a root of a
composite field.
18. The medium of claim 17 further storing instructions to ensure
that an element exists in the field such that none of the powers of
the element is one.
19. An apparatus comprising: a memory write path to use a first set
of polynomials; and a memory read path to use a different set of
polynomials.
20. The apparatus of claim 19 said apparatus to use encryption
operations for reading.
21. The apparatus of claim 19 said apparatus to use Advanced
Encryption Standard.
22. The apparatus of claim 19 said apparatus to select polynomials
to optimize area usage.
23. The apparatus of claim 19 said apparatus to select polynomials
to optimize power consumption.
24. The apparatus of claim 19 said apparatus to use Galois
polynomials.
25. The apparatus of claim 19 said apparatus to use irreducible
polynomials.
26. The apparatus of claim 19 said apparatus to locate a primitive
element that is both a generator and a root of a composite
field.
27. The apparatus of claim 26 said apparatus to ensure that an
element exists in the field such that none of the powers of the
element is one.
28. A system comprising: a core; a memory coupled to the core; a
memory encryption engine coupled to said core, said engine to use
in first set of polynomials for encryption and a different set of
polynomials for decryption; and a network interface card coupled to
said core.
29. The system of claim 28, said engine to use encryption
operations for reading.
30. The system of claim 19, said engine to use irreducible
polynomials.
Description
BACKGROUND
[0001] This relates generally to a memory encryption engine.
[0002] A memory encryption engine is used to protect data as it is
written to and read from memory. Typically the encryption uses the
Advanced Encryption Standard (AES). See NIST Advanced Encryption
Standard (FIP pub. 197, Nov. 26, 2001). The Advanced Encryption
Standard is a symmetric-key encryption protocol used to encrypt and
decrypt all read and write memory accesses. In order to prevent
reads and writes from swamping processor performance, hardware
accelerated AES encrypt and decrypt operations are desirable.
[0003] AES provides several modes of operations. AES-128, AES-192
and AES-256 modes of operation submit 128-bit input data to
respectfully, ten, twelve, and fourteen iterations of an AES round
operation. The AES round operation includes successive Substitute
Bytes, ShiftRow and MixedColumns transformations, followed by an
AddRoundKey operation.
[0004] During the Substitute Bytes transformation, each 8-bits of
the 128-bit input data is input into one of sixteen S-boxes. Each
S-box computes the multiplicative inverse of its respective 8-bit
input in the Galois Field GF(2.sup.8). Some implementations map the
8-bit input to a composite field, (GF(2.sup.4).sup.2), compute the
multiplicative inverse in GF(2.sup.4).sup.2, map the result back to
a ground field GF(2.sup.8), and proceed to the shift row
transformation.
BRIEF DESCRIPTION OF THE DRAWINGS
[0005] Some embodiments are described with respect to the following
figures:
[0006] FIG. 1 is a schematic depiction of a memory encryption
engine;
[0007] FIG. 2 is an advanced encryption standard S-box according to
one embodiment;
[0008] FIG. 3 is a depiction of the multiplier equations according
to one embodiment to the present invention;
[0009] FIG. 4 is a depiction of the GF(2.sup.4) multiplier
according to one embodiment;
[0010] FIG. 5 is a depiction of an S-box subblock for encrypt and
decrypt according to one embodiment;
[0011] FIG. 6 is a depiction of an S-box subblock for encrypt and
decrypt according to another embodiment;
[0012] FIG. 7 is a schematic depiction of MixColumn block for
encrypt according to one embodiment;
[0013] FIG. 8 is a flow chart for one embodiment; and
[0014] FIG. 9 is a system depiction for one embodiment.
DETAILED DESCRIPTION
[0015] In accordance with some embodiments, different sets of
polynomials are selected for encryption and decryption
accelerators. That is, different sets of polynomials are used for
encryption and decryption, each set being chosen to use less area
and deliver more power for a memory encryption engine. This is
advantageous in some embodiments since memory read operations are
typically more critical and latency sensitive than memory
writes.
[0016] Referring to FIG. 1, read data from memory 26 is provided to
a two to one multiplexer in a memory encryption engine 10 and then
to an AddRoundKey unit 14 in the memory read path. From there the
data goes to a Substitute Bytes block 16, ShiftRows block 18 and
MixColumns/AddRoundKey block 20. After ten iterations, according to
one embodiment, the read data from the core 22 is output. The core
22 may be a processor such as a central processing unit.
[0017] Work data from the core 22 is provided to a two to one
multiplexer in the memory write path and then to an inverse
MixColumn/AddRoundKey unit 20a. From here the data goes to an
inverse Substitute Bytes unit 16a and an InverseShiftRows unit 18a.
Finally the data is outputted from an AddRoundKey unit 14a of write
data to memory 26 after ten iterations, according to one
embodiment.
[0018] In some embodiments, a trade-off is made to improve a read
path by using simpler computations of AES-128 encrypt during a
memory read, while using AES-128 decrypt during memory writes. This
avoids using more complex AES-128 decrypt memory reads. The
presence of a larger number of read ports compared to write ports
also makes this trade-off attractive from a silicon area use
perspective.
[0019] The use of separate encrypt and decrypt hardware for
simultaneous read and write operations makes the use of the same
set of polynomials for both encrypt and decrypt suboptimal. Thus
some embodiments use two sets of polynomials: one for encrypt and
the other for decrypt.
[0020] To facilitate inverse computation in the Substitute Bytes,
the plaintext operands in GF(2.sup.8) are mapped to the
composite-field of GF(2.sup.4).sup.2. The corresponding two-term
element in the composite field is represented as shx+sl, where the
elements sh and sl are terms in the field of GF(2.sup.4) and the
composite-field is defined by the polynomial
x.sup.2+.alpha.x+.beta.. Operations in the ground field of
GF(2.sup.4) are, on the other hand, defined by a ground-field
polynomial. There are sixteen potential choices for the
ground-field polynomial of order four, ranging from x.sup.4,
x.sup.4+1 . . . x.sup.4+x.sup.3+x.sup.2+x+1. The ground-field
polynomial is a polynomial that is irreducible over GF(2), i.e. it
does not have a root in GF(2)={0,1}. This requirement eliminates
most choices, leaving x.sup.4+x+1, x.sup.4+x.sup.3+1 and
x.sup.4+x.sup.3+x.sup.2+x+1 as potential ground-field
polynomials.
[0021] The composite-field GF(2.sup.4).sup.2 is an extension of the
ground field GF(2.sup.4). It is therefore associated with a
generator polynomial known as the composite-field polynomial
x.sup.2+.alpha.x+.beta., where .alpha. and .beta. are elements of
GF(2.sup.4). In some embodiments the polynomial may be irreducible
(i.e. not have a root) in GF(2.sup.4). There are 256 potential
candidates for the composite-field polynomial, ranging from
x.sup.2, x.sup.2+1, . . . x.sup.2+Fx+E, x.sup.2+Fx+F. The list of
4096 possible combinations of ground and composite-field
polynomials is pruned down to 360 combinations by the test for
irreducibility. The next step involves the search for an element
`e` in GF(2.sup.4).sup.2 that is both a root of the composite-field
(i.e e.sup.2+.alpha.e+.beta.=0) and has some power `y` that is also
a root of the original GF(2.sup.8) generator polynomial (i.e.
(e.sup.y).sup.8+(e.sup.y).sup.4+(e.sup.y).sup.3+(e.sup.y)+1=0). The
element e.sup.y forms the basis of the composite-field. The above
tests yields eight potential bases in each of the 360 combinations,
leading to 2880 valid representations for the composite-field.
TABLE-US-00001 Ground-Field Poly = x.sup.4 + x + 1 C-F poly Basis
C-F poly Basis C-F poly Basis C-F poly Basis C-F poly Basis C-F
poly Basis C-F poly Basis C-F poly Basis x.sup.2 + 1x + 8 20
x.sup.2 + 2x + 1 1D x.sup.2 + 3x + 1 18 x.sup.2 + 4x + 1 1A x.sup.2
+ 5x + 1 19 x.sup.2 + 6x + 2 8A x.sup.2 + 7x + 2 A9 x.sup.2 + 8x +
1 21 x.sup.2 + 1x + 8 22 x.sup.2 + 2x + 1 1F x.sup.2 + 3x + 1 1B
x.sup.2 + 4x + 1 1E x.sup.2 + 5x + 1 1C x.sup.2 + 6x + 2 8F x.sup.2
+ 7x + 2 AA x.sup.2 + 8x + 1 22 x.sup.2 + 1x + 8 3C x.sup.2 + 2x +
1 2B x.sup.2 + 3x + 1 38 x.sup.2 + 4x + 1 49 x.sup.2 + 5x + 1 5C
x.sup.2 + 6x + 2 94 x.sup.2 + 7x + 2 BA x.sup.2 + 8x + 1 60 x.sup.2
+ 1x + 8 3F x.sup.2 + 2x + 1 2F x.sup.2 + 3x + 1 3D x.sup.2 + 4x +
1 4A x.sup.2 + 5x + 1 5E x.sup.2 + 6x + 2 97 x.sup.2 + 7x + 2 BE
x.sup.2 + 8x + 1 65 x.sup.2 + 1x + 8 42 x.sup.2 + 2x + 1 8D x.sup.2
+ 3x + 1 D3 x.sup.2 + 4x + 1 94 x.sup.2 + 5x + 1 A9 x.sup.2 + 6x +
2 E1 x.sup.2 + 7x + 2 C4 x.sup.2 + 8x + 1 98 x.sup.2 + 1x + 8 46
x.sup.2 + 2x + 1 8E x.sup.2 + 3x + 1 D7 x.sup.2 + 4x + 1 96 x.sup.2
+ 5x + 1 AD x.sup.2 + 6x + 2 E3 x.sup.2 + 7x + 2 C6 x.sup.2 + 8x +
1 9C x.sup.2 + 1x + 8 51 x.sup.2 + 2x + 1 B2 x.sup.2 + 3x + 1 F9
x.sup.2 + 4x + 1 CB x.sup.2 + 5x + 1 E5 x.sup.2 + 6x + 2 F0 x.sup.2
+ 7x + 2 DB x.sup.2 + 8x + 1 D5 x.sup.2 + 1x + 8 54 x.sup.2 + 2x +
1 B7 x.sup.2 + 3x + 1 FB x.sup.2 + 4x + 1 CE x.sup.2 + 5x + 1 E6
x.sup.2 + 6x + 2 F4 x.sup.2 + 7x + 2 DE x.sup.2 + 8x + 1 D7 x.sup.2
+ 1x + 9 2C x.sup.2 + 2x + 2 1C x.sup.2 + 3x + 3 19 x.sup.2 + 4x +
2 11 x.sup.2 + 5x + 3 11 x.sup.2 + 6x + 3 80 x.sup.2 + 7x + 3 A5
x.sup.2 + 8x + 3 2D x.sup.2 + 1x + 9 2E x.sup.2 + 2x + 2 1E x.sup.2
+ 3x + 3 1A x.sup.2 + 4x + 2 15 x.sup.2 + 5x + 3 14 x.sup.2 + 6x +
3 85 x.sup.2 + 7x + 3 A6 x.sup.2 + 8x + 3 2E x.sup.2 + 1x + 9 35
x.sup.2 + 2x + 2 29 x.sup.2 + 3x + 3 3B x.sup.2 + 4x + 2 40 x.sup.2
+ 5x + 3 50 x.sup.2 + 6x + 3 91 x.sup.2 + 7x + 3 B9 x.sup.2 + 8x +
3 62 x.sup.2 + 1x + 9 36 x.sup.2 + 2x + 2 2D x.sup.2 + 3x + 3 3E
x.sup.2 + 4x + 2 43 x.sup.2 + 5x + 3 52 x.sup.2 + 6x + 3 92 x.sup.2
+ 7x + 3 BD x.sup.2 + 8x + 3 67 x.sup.2 + 1x + 9 49 x.sup.2 + 2x +
2 85 x.sup.2 + 3x + 3 DA x.sup.2 + 4x + 2 98 x.sup.2 + 5x + 3 A2
x.sup.2 + 6x + 3 E5 x.sup.2 + 7x + 3 CC x.sup.2 + 8x + 3 9B x.sup.2
+ 1x + 9 4D x.sup.2 + 2x + 2 86 x.sup.2 + 3x + 3 DE x.sup.2 + 4x +
2 9A x.sup.2 + 5x + 3 A6 x.sup.2 + 6x + 3 E7 x.sup.2 + 7x + 3 CE
x.sup.2 + 8x + 3 9F x.sup.2 + 1x + 9 59 x.sup.2 + 2x + 2 B9 x.sup.2
+ 3x + 3 F4 x.sup.2 + 4x + 2 C3 x.sup.2 + 5x + 3 EC x.sup.2 + 6x +
3 F8 x.sup.2 + 7x + 3 D9 x.sup.2 + 8x + 3 DD x.sup.2 + 1x + 9 5C
x.sup.2 + 2x + 2 BC x.sup.2 + 3x + 3 F6 x.sup.2 + 4x + 2 C6 x.sup.2
+ 5x + 3 EF x.sup.2 + 6x + 3 FC x.sup.2 + 7x + 3 DC x.sup.2 + 8x +
3 DF x.sup.2 + 1x + A 25 x.sup.2 + 2x + 5 11 x.sup.2 + 3x + 4 11
x.sup.2 + 4x + 4 1B x.sup.2 + 5x + 5 18 x.sup.2 + 6x + 4 88 x.sup.2
+ 7x + 4 A0 x.sup.2 + 8x + 4 29 x.sup.2 + 1x + A 27 x.sup.2 + 2x +
5 13 x.sup.2 + 3x + 4 12 x.sup.2 + 4x + 4 1F x.sup.2 + 5x + 5 1D
x.sup.2 + 6x + 4 8D x.sup.2 + 7x + 4 A3 x.sup.2 + 8x + 4 2A x.sup.2
+ 1x + A 31 x.sup.2 + 2x + 5 20 x.sup.2 + 3x + 4 30 x.sup.2 + 4x +
4 4D x.sup.2 + 5x + 5 59 x.sup.2 + 6x + 4 98 x.sup.2 + 7x + 4 B1
x.sup.2 + 8x + 4 6B x.sup.2 + 1x + A 32 x.sup.2 + 2x + 5 24 x.sup.2
+ 3x + 4 35 x.sup.2 + 4x + 4 4E x.sup.2 + 5x + 5 5B x.sup.2 + 6x +
4 9B x.sup.2 + 7x + 4 B5 x.sup.2 + 8x + 4 6E x.sup.2 + 1x + A 48
x.sup.2 + 2x + 5 84 x.sup.2 + 3x + 4 D8 x.sup.2 + 4x + 4 9D x.sup.2
+ 5x + 5 A3 x.sup.2 + 6x + 4 E9 x.sup.2 + 7x + 4 C8 x.sup.2 + 8x +
4 9A x.sup.2 + 1x + A 4C x.sup.2 + 2x + 5 87 x.sup.2 + 3x + 4 DC
x.sup.2 + 4x + 4 9F x.sup.2 + 5x + 5 A7 x.sup.2 + 6x + 4 EB x.sup.2
+ 7x + 4 CA x.sup.2 + 8x + 4 9E x.sup.2 + 1x + A 50 x.sup.2 + 2x +
5 BA x.sup.2 + 3x + 4 F5 x.sup.2 + 4x + 4 C2 x.sup.2 + 5x + 5 E8
x.sup.2 + 6x + 4 F3 x.sup.2 + 7x + 4 D3 x.sup.2 + 8x + 4 D4 x.sup.2
+ 1x + A 55 x.sup.2 + 2x + 5 BF x.sup.2 + 3x + 4 F7 x.sup.2 + 4x +
4 C7 x.sup.2 + 5x + 5 EB x.sup.2 + 6x + 4 F7 x.sup.2 + 7x + 4 D6
x.sup.2 + 8x + 4 D6 x.sup.2 + 1x + B 29 x.sup.2 + 2x + 6 10 x.sup.2
+ 3x + 6 10 x.sup.2 + 4x + 7 10 x.sup.2 + 5x + 7 10 x.sup.2 + 6x +
5 82 x.sup.2 + 7x + 5 AC x.sup.2 + 8x + 6 25 x.sup.2 + 1x + B 2B
x.sup.2 + 2x + 6 12 x.sup.2 + 3x + 6 13 x.sup.2 + 4x + 7 14 x.sup.2
+ 5x + 7 15 x.sup.2 + 6x + 5 87 x.sup.2 + 7x + 5 AF x.sup.2 + 8x +
6 26 x.sup.2 + 1x + B 38 x.sup.2 + 2x + 6 22 x.sup.2 + 3x + 6 33
x.sup.2 + 4x + 7 44 x.sup.2 + 5x + 7 55 x.sup.2 + 6x + 5 9D x.sup.2
+ 7x + 5 B2 x.sup.2 + 8x + 6 69 x.sup.2 + 1x + B 3B x.sup.2 + 2x +
6 26 x.sup.2 + 3x + 6 36 x.sup.2 + 4x + 7 47 x.sup.2 + 5x + 7 57
x.sup.2 + 6x + 5 9E x.sup.2 + 7x + 5 B6 x.sup.2 + 8x + 6 6C x.sup.2
+ 1x + B 43 x.sup.2 + 2x + 6 8C x.sup.2 + 3x + 6 D1 x.sup.2 + 4x +
7 91 x.sup.2 + 5x + 7 A8 x.sup.2 + 6x + 5 ED x.sup.2 + 7x + 5 C0
x.sup.2 + 8x + 6 99 x.sup.2 + 1x + B 47 x.sup.2 + 2x + 6 8F x.sup.2
+ 3x + 6 D5 x.sup.2 + 4x + 7 93 x.sup.2 + 5x + 7 AC x.sup.2 + 6x +
5 EF x.sup.2 + 7x + 5 C2 x.sup.2 + 8x + 6 9D x.sup.2 + 1x + B 58
x.sup.2 + 2x + 6 B1 x.sup.2 + 3x + 6 F8 x.sup.2 + 4x + 7 CA x.sup.2
+ 5x + 7 E1 x.sup.2 + 6x + 5 FB x.sup.2 + 7x + 5 D1 x.sup.2 + 8x +
6 DC x.sup.2 + 1x + B 5D x.sup.2 + 2x + 6 B4 x.sup.2 + 3x + 6 FA
x.sup.2 + 4x + 7 CF x.sup.2 + 5x + 7 E2 x.sup.2 + 6x + 5 FF x.sup.2
+ 7x + 5 D4 x.sup.2 + 8x + 6 DE x.sup.2 + 1x + C 21 x.sup.2 + 2x +
9 18 x.sup.2 + 3x + 9 14 x.sup.2 + 4x + 8 19 x.sup.2 + 5x + 9 12
x.sup.2 + 6x + A 89 x.sup.2 + 7x + 8 AD x.sup.2 + 8x + 8 20 x.sup.2
+ 1x + C 23 x.sup.2 + 2x + 9 1A x.sup.2 + 3x + 9 17 x.sup.2 + 4x +
8 1D x.sup.2 + 5x + 9 17 x.sup.2 + 6x + A 8C x.sup.2 + 7x + 8 AE
x.sup.2 + 8x + 8 23 x.sup.2 + 1x + C 34 x.sup.2 + 2x + 9 21 x.sup.2
+ 3x + 9 3A x.sup.2 + 4x + 8 45 x.sup.2 + 5x + 9 5D x.sup.2 + 6x +
A 95 x.sup.2 + 7x + 8 BB x.sup.2 + 8x + 8 63 x.sup.2 + 1x + C 37
x.sup.2 + 2x + 9 25 x.sup.2 + 3x + 9 3F x.sup.2 + 4x + 8 46 x.sup.2
+ 5x + 9 5F x.sup.2 + 6x + A 96 x.sup.2 + 7x + 8 BF x.sup.2 + 8x +
8 66 x.sup.2 + 1x + C 40 x.sup.2 + 2x + 9 80 x.sup.2 + 3x + 9 D0
x.sup.2 + 4x + 8 9C x.sup.2 + 5x + 9 AB x.sup.2 + 6x + A EC x.sup.2
+ 7x + 8 CD x.sup.2 + 8x + 8 91 x.sup.2 + 1x + C 44 x.sup.2 + 2x +
9 83 x.sup.2 + 3x + 9 D4 x.sup.2 + 4x + 8 9E x.sup.2 + 5x + 9 AF
x.sup.2 + 6x + A EE x.sup.2 + 7x + 8 CF x.sup.2 + 8x + 8 95 x.sup.2
+ 1x + C 5A x.sup.2 + 2x + 9 B3 x.sup.2 + 3x + 9 F1 x.sup.2 + 4x +
8 C9 x.sup.2 + 5x + 9 ED x.sup.2 + 6x + A F9 x.sup.2 + 7x + 8 D2
x.sup.2 + 8x + 8 D8 x.sup.2 + 1x + C 5F x.sup.2 + 2x + 9 B6 x.sup.2
+ 3x + 9 F3 x.sup.2 + 4x + 8 CC x.sup.2 + 5x + 9 EE x.sup.2 + 6x +
A FD x.sup.2 + 7x + 8 D7 x.sup.2 + 8x + 8 DA x.sup.2 + 1x + D 2D
x.sup.2 + 2x + A 19 x.sup.2 + 3x + B 15 x.sup.2 + 4x + B 12 x.sup.2
+ 5x + B 1A x.sup.2 + 6x + B 83 x.sup.2 + 7x + 9 A1 x.sup.2 + 8x +
A 2C x.sup.2 + 1x + D 2F x.sup.2 + 2x + A 1B x.sup.2 + 3x + B 16
x.sup.2 + 4x + B 16 x.sup.2 + 5x + B 1F x.sup.2 + 6x + B 86 x.sup.2
+ 7x + 9 A2 x.sup.2 + 8x + A 2F x.sup.2 + 1x + D 3D x.sup.2 + 2x +
A 23 x.sup.2 + 3x + B 39 x.sup.2 + 4x + B 4C x.sup.2 + 5x + B 51
x.sup.2 + 6x + B 90 x.sup.2 + 7x + 9 B8 x.sup.2 + 8x + A 61 x.sup.2
+ 1x + D 3E x.sup.2 + 2x + A 27 x.sup.2 + 3x + B 3C x.sup.2 + 4x +
B 4F x.sup.2 + 5x + B 53 x.sup.2 + 6x + B 93 x.sup.2 + 7x + 9 BC
x.sup.2 + 8x + A 64 x.sup.2 + 1x + D 4B x.sup.2 + 2x + A 88 x.sup.2
+ 3x + B D9 x.sup.2 + 4x + B 90 x.sup.2 + 5x + B A0 x.sup.2 + 6x +
B E8 x.sup.2 + 7x + 9 C5 x.sup.2 + 8x + A 92 x.sup.2 + 1x + D 4F
x.sup.2 + 2x + A 8B x.sup.2 + 3x + B DD x.sup.2 + 4x + B 92 x.sup.2
+ 5x + B A4 x.sup.2 + 6x + B EA x.sup.2 + 7x + 9 C7 x.sup.2 + 8x +
A 96 x.sup.2 + 1x + D 52 x.sup.2 + 2x + A B8 x.sup.2 + 3x + B FC
x.sup.2 + 4x + B C1 x.sup.2 + 5x + B E4 x.sup.2 + 6x + B F1 x.sup.2
+ 7x + 9 D0 x.sup.2 + 8x + A D0 x.sup.2 + 1x + D 57 x.sup.2 + 2x +
A BD x.sup.2 + 3x + B FE x.sup.2 + 4x + B C4 x.sup.2 + 5x + B E7
x.sup.2 + 6x + B F5 x.sup.2 + 7x + 9 D5 x.sup.2 + 8x + A D2 x.sup.2
+ 1x + E 24 x.sup.2 + 2x + D 14 x.sup.2 + 3x + C 1D x.sup.2 + 4x +
D 18 x.sup.2 + 5x + D 13 x.sup.2 + 6x + C 8B x.sup.2 + 7x + E A4
x.sup.2 + 8x + D 28 x.sup.2 + 1x + E 26 x.sup.2 + 2x + D 16 x.sup.2
+ 3x + C 1E x.sup.2 + 4x + D 1C x.sup.2 + 5x + D 16 x.sup.2 + 6x +
C 8E x.sup.2 + 7x + E A7 x.sup.2 + 8x + D 2B x.sup.2 + 1x + E 39
x.sup.2 + 2x + D 2A x.sup.2 + 3x + C 32 x.sup.2 + 4x + D 41 x.sup.2
+ 5x + D 58 x.sup.2 + 6x + C 99 x.sup.2 + 7x + E B0 x.sup.2 + 8x +
D 68 x.sup.2 + 1x + E 3A x.sup.2 + 2x + D 2E x.sup.2 + 3x + C 37
x.sup.2 + 4x + D 42 x.sup.2 + 5x + D 5A x.sup.2 + 6x + C 9A x.sup.2
+ 7x + E B4 x.sup.2 + 8x + D 6D x.sup.2 + 1x + E 4A x.sup.2 + 2x +
D 89 x.sup.2 + 3x + C DB x.sup.2 + 4x + D 95 x.sup.2 + 5x + D A1
x.sup.2 + 6x + C E4 x.sup.2 + 7x + E C1 x.sup.2 + 8x + D 93 x.sup.2
+ 1x + E 4E x.sup.2 + 2x + D 8A x.sup.2 + 3x + C DF x.sup.2 + 4x +
D 97 x.sup.2 + 5x + D A5 x.sup.2 + 6x + C E6 x.sup.2 + 7x + E C3
x.sup.2 + 8x + D 97 x.sup.2 + 1x + E 5B x.sup.2 + 2x + D BB x.sup.2
+ 3x + C FD x.sup.2 + 4x + D C0 x.sup.2 + 5x + D E0 x.sup.2 + 6x +
C FA x.sup.2 + 7x + E DA x.sup.2 + 8x + D D9 x.sup.2 + 1x + E 5E
x.sup.2 + 2x + D BE x.sup.2 + 3x + C FF x.sup.2 + 4x + D C5 x.sup.2
+ 5x + D E3 x.sup.2 + 6x + C FE x.sup.2 + 7x + E DF x.sup.2 + 8x +
D DB x.sup.2 + 1x + F 28 x.sup.2 + 2x + E 15 x.sup.2 + 3x + E 1C
x.sup.2 + 4x + E 13 x.sup.2 + 5x + F 1B x.sup.2 + 6x + D 81 x.sup.2
+ 7x + F A8 x.sup.2 + 8x + F 24 x.sup.2 + 1x + F 2A x.sup.2 + 2x +
E 17 x.sup.2 + 3x + E 1F x.sup.2 + 4x + E 17 x.sup.2 + 5x + F 1E
x.sup.2 + 6x + D 84 x.sup.2 + 7x + F AB x.sup.2 + 8x + F 27 x.sup.2
+ 1x + F 30 x.sup.2 + 2x + E 28 x.sup.2 + 3x + E 31 x.sup.2 + 4x +
E 48 x.sup.2 + 5x + F 54 x.sup.2 + 6x + D 9C x.sup.2 + 7x + F B3
x.sup.2 + 8x + F 6A x.sup.2 + 1x + F 33 x.sup.2 + 2x + E 2C x.sup.2
+ 3x + E 34 x.sup.2 + 4x + E 4B x.sup.2 + 5x + F 56 x.sup.2 + 6x +
D 9F x.sup.2 + 7x + F B7 x.sup.2 + 8x + F 6F x.sup.2 + 1x + F 41
x.sup.2 + 2x + E 81 x.sup.2 + 3x + E D2 x.sup.2 + 4x + E 99 x.sup.2
+ 5x + F AA x.sup.2 + 6x + D E0 x.sup.2 + 7x + F C9 x.sup.2 + 8x +
F 90 x.sup.2 + 1x + F 45 x.sup.2 + 2x + E 82 x.sup.2 + 3x + E D6
x.sup.2 + 4x + E 9B x.sup.2 + 5x + F AE x.sup.2 + 6x + D E2 x.sup.2
+ 7x + F CB x.sup.2 + 8x + F 94 x.sup.2 + 1x + F 53 x.sup.2 + 2x +
E B0 x.sup.2 + 3x + E F0 x.sup.2 + 4x + E C8 x.sup.2 + 5x + F E9
x.sup.2 + 6x + D F2 x.sup.2 + 7x + F D8 x.sup.2 + 8x + F D1 x.sup.2
+ 1x + F 56 x.sup.2 + 2x + E B5 x.sup.2 + 3x + E F2 x.sup.2 + 4x +
E CD x.sup.2 + 5x + F EA x.sup.2 + 6x + D F6 x.sup.2 + 7x + F DD
x.sup.2 + 8x + F D3 Ground-Field Poly = x.sup.4 + x + 1 C-F poly
Basis C-F poly Basis C-F poly Basis C-F poly Basis C-F poly Basis
C-F poly Basis C-F poly Basis x.sup.2 + 9x + 2 40 x.sup.2 + Ax + 1
51 x.sup.2 + Bx + 4 21 x.sup.2 + Cx + 1 41 x.sup.2 + Dx + 4 30
x.sup.2 + Ex + 2 51 x.sup.2 + Fx + 1 31 x.sup.2 + 9x + 2 42 x.sup.2
+ Ax + 1 55 x.sup.2 + Bx + 4 24 x.sup.2 + Cx + 1 44 x.sup.2 + Dx +
4 34 x.sup.2 + Ex + 2 52 x.sup.2 + Fx + 1 33 x.sup.2 + 9x + 2 6C
x.sup.2 + Ax + 1 70 x.sup.2 + Bx + 4 72 x.sup.2 + Cx + 1 70 x.sup.2
+ Dx + 4 7A x.sup.2 + Ex + 2 65 x.sup.2 + Fx + 1 60 x.sup.2 + 9x +
2 6F x.sup.2 + Ax + 1 73 x.sup.2 + Bx + 4 76 x.sup.2 + Cx + 1 72
x.sup.2 + Dx + 4 7F x.sup.2 + Ex + 2 67 x.sup.2 + Fx + 1 64 x.sup.2
+ 9x + 2 82 x.sup.2 + Ax + 1 93 x.sup.2 + Bx + 4 A0 x.sup.2 + Cx +
1 DC x.sup.2 + Dx + 4 81 x.sup.2 + Ex + 2 C8 x.sup.2 + Fx + 1 B4
x.sup.2 + 9x + 2 86 x.sup.2 + Ax + 1 96 x.sup.2 + Bx + 4 A2 x.sup.2
+ Cx + 1 DF x.sup.2 + Dx + 4 83 x.sup.2 + Ex + 2 CC x.sup.2 + Fx +
1 B7 x.sup.2 + 9x + 2 A1 x.sup.2 + Ax + 1 B8 x.sup.2 + Bx + 4 FC
x.sup.2 + Cx + 1 E2 x.sup.2 + Dx + 4 C4 x.sup.2 + Ex + 2 F0 x.sup.2
+ Fx + 1 EA x.sup.2 + 9x + 2 A4 x.sup.2 + Ax + 1 BA x.sup.2 + Bx +
4 FF x.sup.2 + Cx + 1 E6 x.sup.2 + Dx + 4 C7 x.sup.2 + Ex + 2 F5
x.sup.2 + Fx + 1 EF x.sup.2 + 9x + 3 41 x.sup.2 + Ax + 2 5B x.sup.2
+ Bx + 5 20 x.sup.2 + Cx + 3 48 x.sup.2 + Dx + 5 31 x.sup.2 + Ex +
3 50 x.sup.2 + Fx + 2 39 x.sup.2 + 9x + 3 43 x.sup.2 + Ax + 2 5F
x.sup.2 + Bx + 5 25 x.sup.2 + Cx + 3 4D x.sup.2 + Dx + 5 35 x.sup.2
+ Ex + 3 53 x.sup.2 + Fx + 2 3B x.sup.2 + 9x + 3 64 x.sup.2 + Ax +
2 7D x.sup.2 + Bx + 5 78 x.sup.2 + Cx + 3 79 x.sup.2 + Dx + 5 73
x.sup.2 + Ex + 3 68 x.sup.2 + Fx + 2 63 x.sup.2 + 9x + 3 67 x.sup.2
+ Ax + 2 7E x.sup.2 + Bx + 5 7C x.sup.2 + Cx + 3 7B x.sup.2 + Dx +
5 76 x.sup.2 + Ex + 3 6A x.sup.2 + Fx + 2 67 x.sup.2 + 9x + 3 80
x.sup.2 + Ax + 2 92 x.sup.2 + Bx + 5 A5 x.sup.2 + Cx + 3 D8 x.sup.2
+ Dx + 5 88 x.sup.2 + Ex + 3 C1 x.sup.2 + Fx + 2 B8 x.sup.2 + 9x +
3 84 x.sup.2 + Ax + 2 97 x.sup.2 + Bx + 5 A7 x.sup.2 + Cx + 3 DB
x.sup.2 + Dx + 5 8A x.sup.2 + Ex + 3 C5 x.sup.2 + Fx + 2 BB x.sup.2
+ 9x + 3 AA x.sup.2 + Ax + 2 BD x.sup.2 + Bx + 5 F1 x.sup.2 + Cx +
3 E3 x.sup.2 + Dx + 5 C0 x.sup.2 + Ex + 3 F3 x.sup.2 + Fx + 2 E8
x.sup.2 + 9x + 3 AF x.sup.2 + Ax + 2 BF x.sup.2 + Bx + 5 F2 x.sup.2
+ Cx + 3 E7 x.sup.2 + Dx + 5 C3 x.sup.2 + Ex + 3 F6 x.sup.2 + Fx +
2 ED x.sup.2 + 9x + 6 49 x.sup.2 + Ax + 4 59 x.sup.2 + Bx + 6 2A
x.sup.2 + Cx + 5 4B x.sup.2 + Dx + 6 38 x.sup.2 + Ex + 6 5D x.sup.2
+ Fx + 5 3C x.sup.2 + 9x + 6 4B x.sup.2 + Ax + 4 5D x.sup.2 + Bx +
6 2F x.sup.2 + Cx + 5 4E x.sup.2 + Dx + 6 3C x.sup.2 + Ex + 6 5E
x.sup.2 + Fx + 5 3E x.sup.2 + 9x + 6 68 x.sup.2 + Ax + 4 75 x.sup.2
+ Bx + 6 70 x.sup.2 + Cx + 5 74 x.sup.2 + Dx + 6 70 x.sup.2 + Ex +
6 6D x.sup.2 + Fx + 5 69 x.sup.2 + 9x + 6 6B x.sup.2 + Ax + 4 76
x.sup.2 + Bx + 6 74 x.sup.2 + Cx + 5 76 x.sup.2 + Dx + 6 75 x.sup.2
+ Ex + 6 6F x.sup.2 + Fx + 5 6D x.sup.2 + 9x + 6 83 x.sup.2 + Ax +
4 99 x.sup.2 + Bx + 6 A1 x.sup.2 + Cx + 5 D9 x.sup.2 + Dx + 6 85
x.sup.2 + Ex + 6 CB x.sup.2 + Fx + 5 B5 x.sup.2 + 9x + 6 87 x.sup.2
+ Ax + 4 9C x.sup.2 + Bx + 6 A3 x.sup.2 + Cx + 5 DA x.sup.2 + Dx +
6 87 x.sup.2 + Ex + 6 CF x.sup.2 + Fx + 5 B6 x.sup.2 + 9x + 6 A8
x.sup.2 + Ax + 4 BC x.sup.2 + Bx + 6 F4 x.sup.2 + Cx + 5 EA x.sup.2
+ Dx + 6 C1 x.sup.2 + Ex + 6 F2 x.sup.2 + Fx + 5 E9 x.sup.2 + 9x +
6 AD x.sup.2 + Ax + 4 BE x.sup.2 + Bx + 6 F7 x.sup.2 + Cx + 5 EE
x.sup.2 + Dx + 6 C2 x.sup.2 + Ex + 6 F7 x.sup.2 + Fx + 5 EC
x.sup.2 + 9x + 7 48 x.sup.2 + Ax + 7 53 x.sup.2 + Bx + 7 2B x.sup.2
+ Cx + 7 42 x.sup.2 + Dx + 7 39 x.sup.2 + Ex + 7 5C x.sup.2 + Fx +
6 34 x.sup.2 + 9x + 7 4A x.sup.2 + Ax + 7 57 x.sup.2 + Bx + 7 2E
x.sup.2 + Cx + 7 47 x.sup.2 + Dx + 7 3D x.sup.2 + Ex + 7 5F x.sup.2
+ Fx + 6 36 x.sup.2 + 9x + 7 60 x.sup.2 + Ax + 7 78 x.sup.2 + Bx +
7 7A x.sup.2 + Cx + 7 7D x.sup.2 + Dx + 7 79 x.sup.2 + Ex + 7 60
x.sup.2 + Fx + 6 6A x.sup.2 + 9x + 7 63 x.sup.2 + Ax + 7 7B x.sup.2
+ Bx + 7 7E x.sup.2 + Cx + 7 7F x.sup.2 + Dx + 7 7C x.sup.2 + Ex +
7 62 x.sup.2 + Fx + 6 6E x.sup.2 + 9x + 7 81 x.sup.2 + Ax + 7 98
x.sup.2 + Bx + 7 A4 x.sup.2 + Cx + 7 DD x.sup.2 + Dx + 7 8C x.sup.2
+ Ex + 7 C2 x.sup.2 + Fx + 6 B9 x.sup.2 + 9x + 7 85 x.sup.2 + Ax +
7 9D x.sup.2 + Bx + 7 A6 x.sup.2 + Cx + 7 DE x.sup.2 + Dx + 7 8E
x.sup.2 + Ex + 7 C6 x.sup.2 + Fx + 6 BA x.sup.2 + 9x + 7 A3 x.sup.2
+ Ax + 7 B9 x.sup.2 + Bx + 7 F9 x.sup.2 + Cx + 7 EB x.sup.2 + Dx +
7 C5 x.sup.2 + Ex + 7 F1 x.sup.2 + Fx + 6 EB x.sup.2 + 9x + 7 A6
x.sup.2 + Ax + 7 BB x.sup.2 + Bx + 7 FA x.sup.2 + Cx + 7 EF x.sup.2
+ Dx + 7 C6 x.sup.2 + Ex + 7 F4 x.sup.2 + Fx + 6 EE x.sup.2 + 9x +
A 44 x.sup.2 + Ax + 9 5A x.sup.2 + Bx + C 28 x.sup.2 + Cx + 8 4A
x.sup.2 + Dx + 8 33 x.sup.2 + Ex + 8 59 x.sup.2 + Fx + 8 35 x.sup.2
+ 9x + A 46 x.sup.2 + Ax + 9 5E x.sup.2 + Bx + C 2D x.sup.2 + Cx +
8 4F x.sup.2 + Dx + 8 37 x.sup.2 + Ex + 8 5A x.sup.2 + Fx + 8 37
x.sup.2 + 9x + A 69 x.sup.2 + Ax + 9 79 x.sup.2 + Bx + C 73 x.sup.2
+ Cx + 8 71 x.sup.2 + Dx + 8 78 x.sup.2 + Ex + 8 64 x.sup.2 + Fx +
8 68 x.sup.2 + 9x + A 6A x.sup.2 + Ax + 9 7A x.sup.2 + Bx + C 77
x.sup.2 + Cx + 8 73 x.sup.2 + Dx + 8 7D x.sup.2 + Ex + 8 66 x.sup.2
+ Fx + 8 6C x.sup.2 + 9x + A 8A x.sup.2 + Ax + 9 9B x.sup.2 + Bx +
C A9 x.sup.2 + Cx + 8 D4 x.sup.2 + Dx + 8 89 x.sup.2 + Ex + 8 CA
x.sup.2 + Fx + 8 B1 x.sup.2 + 9x + A 8E x.sup.2 + Ax + 9 9E x.sup.2
+ Bx + C AB x.sup.2 + Cx + 8 D7 x.sup.2 + Dx + 8 8B x.sup.2 + Ex +
8 CE x.sup.2 + Fx + 8 B2 x.sup.2 + 9x + A AB x.sup.2 + Ax + 9 B4
x.sup.2 + Bx + C F8 x.sup.2 + Cx + 8 E0 x.sup.2 + Dx + 8 C8 x.sup.2
+ Ex + 8 FB x.sup.2 + Fx + 8 E0 x.sup.2 + 9x + A AE x.sup.2 + Ax +
9 B6 x.sup.2 + Bx + C FB x.sup.2 + Cx + 8 E4 x.sup.2 + Dx + 8 CB
x.sup.2 + Ex + 8 FE x.sup.2 + Fx + 8 E5 x.sup.2 + 9x + B 45 x.sup.2
+ Ax + A 50 x.sup.2 + Bx + D 29 x.sup.2 + Cx + A 43 x.sup.2 + Dx +
9 32 x.sup.2 + Ex + 9 58 x.sup.2 + Fx + B 3D x.sup.2 + 9x + B 47
x.sup.2 + Ax + A 54 x.sup.2 + Bx + D 2C x.sup.2 + Cx + A 46 x.sup.2
+ Dx + 9 36 x.sup.2 + Ex + 9 5B x.sup.2 + Fx + B 3F x.sup.2 + 9x +
B 61 x.sup.2 + Ax + A 74 x.sup.2 + Bx + D 79 x.sup.2 + Cx + A 78
x.sup.2 + Dx + 9 71 x.sup.2 + Ex + 9 69 x.sup.2 + Fx + B 6B x.sup.2
+ 9x + B 62 x.sup.2 + Ax + A 77 x.sup.2 + Bx + D 7D x.sup.2 + Cx +
A 7A x.sup.2 + Dx + 9 74 x.sup.2 + Ex + 9 6B x.sup.2 + Fx + B 6F
x.sup.2 + 9x + B 88 x.sup.2 + Ax + A 9A x.sup.2 + Bx + D AC x.sup.2
+ Cx + A D0 x.sup.2 + Dx + 9 80 x.sup.2 + Ex + 9 C3 x.sup.2 + Fx +
B BD x.sup.2 + 9x + B 8C x.sup.2 + Ax + A 9F x.sup.2 + Bx + D AE
x.sup.2 + Cx + A D3 x.sup.2 + Dx + 9 82 x.sup.2 + Ex + 9 C7 x.sup.2
+ Fx + B BE x.sup.2 + 9x + B A0 x.sup.2 + Ax + A B1 x.sup.2 + Bx +
D F5 x.sup.2 + Cx + A E1 x.sup.2 + Dx + 9 CC x.sup.2 + Ex + 9 F8
x.sup.2 + Fx + B E2 x.sup.2 + 9x + B A5 x.sup.2 + Ax + A B3 x.sup.2
+ Bx + D F6 x.sup.2 + Cx + A E5 x.sup.2 + Dx + 9 CF x.sup.2 + Ex +
9 FD x.sup.2 + Fx + B E7 x.sup.2 + 9x + E 4D x.sup.2 + Ax + C 52
x.sup.2 + Bx + E 23 x.sup.2 + Cx + C 40 x.sup.2 + Dx + A 3B x.sup.2
+ Ex + C 55 x.sup.2 + Fx + C 38 x.sup.2 + 9x + E 4F x.sup.2 + Ax +
C 56 x.sup.2 + Bx + E 26 x.sup.2 + Cx + C 45 x.sup.2 + Dx + A 3F
x.sup.2 + Ex + C 56 x.sup.2 + Fx + C 3A x.sup.2 + 9x + E 6D x.sup.2
+ Ax + C 7C x.sup.2 + Bx + E 71 x.sup.2 + Cx + C 75 x.sup.2 + Dx +
A 72 x.sup.2 + Ex + C 6C x.sup.2 + Fx + C 61 x.sup.2 + 9x + E 6E
x.sup.2 + Ax + C 7F x.sup.2 + Bx + E 75 x.sup.2 + Cx + C 77 x.sup.2
+ Dx + A 77 x.sup.2 + Ex + C 6E x.sup.2 + Fx + C 65 x.sup.2 + 9x +
E 8B x.sup.2 + Ax + C 91 x.sup.2 + Bx + E A8 x.sup.2 + Cx + C D1
x.sup.2 + Dx + A 8D x.sup.2 + Ex + C C9 x.sup.2 + Fx + C B0 x.sup.2
+ 9x + E 8F x.sup.2 + Ax + C 94 x.sup.2 + Bx + E AA x.sup.2 + Cx +
C D2 x.sup.2 + Dx + A 8F x.sup.2 + Ex + C CD x.sup.2 + Fx + C B3
x.sup.2 + 9x + E A2 x.sup.2 + Ax + C B0 x.sup.2 + Bx + E F0 x.sup.2
+ Cx + C E8 x.sup.2 + Dx + A CD x.sup.2 + Ex + C F9 x.sup.2 + Fx +
C E3 x.sup.2 + 9x + E A7 x.sup.2 + Ax + C B2 x.sup.2 + Bx + E F3
x.sup.2 + Cx + C EC x.sup.2 + Dx + A CE x.sup.2 + Ex + C FC x.sup.2
+ Fx + C E6 x.sup.2 + 9x + F 4C x.sup.2 + Ax + F 58 x.sup.2 + Bx +
F 22 x.sup.2 + Cx + E 49 x.sup.2 + Dx + B 3A x.sup.2 + Ex + D 54
x.sup.2 + Fx + F 30 x.sup.2 + 9x + F 4E x.sup.2 + Ax + F 5C x.sup.2
+ Bx + F 27 x.sup.2 + Cx + E 4C x.sup.2 + Dx + B 3E x.sup.2 + Ex +
D 57 x.sup.2 + Fx + F 32 x.sup.2 + 9x + F 65 x.sup.2 + Ax + F 71
x.sup.2 + Bx + F 7B x.sup.2 + Cx + E 7C x.sup.2 + Dx + B 7B x.sup.2
+ Ex + D 61 x.sup.2 + Fx + F 62 x.sup.2 + 9x + F 66 x.sup.2 + Ax +
F 72 x.sup.2 + Bx + F 7F x.sup.2 + Cx + E 7E x.sup.2 + Dx + B 7E
x.sup.2 + Ex + D 63 x.sup.2 + Fx + F 66 x.sup.2 + 9x + F 89 x.sup.2
+ Ax + F 90 x.sup.2 + Bx + F AD x.sup.2 + Cx + E D5 x.sup.2 + Dx +
B 84 x.sup.2 + Ex + D C0 x.sup.2 + Fx + F BC x.sup.2 + 9x + F 8D
x.sup.2 + Ax + F 95 x.sup.2 + Bx + F AF x.sup.2 + Cx + E D6 x.sup.2
+ Dx + B 86 x.sup.2 + Ex + D C4 x.sup.2 + Fx + F BF x.sup.2 + 9x +
F A9 x.sup.2 + Ax + F B5 x.sup.2 + Bx + F FD x.sup.2 + Cx + E E9
x.sup.2 + Dx + B C9 x.sup.2 + Ex + D FA x.sup.2 + Fx + F E1 x.sup.2
+ 9x + F AC x.sup.2 + Ax + F B7 x.sup.2 + Bx + F FE x.sup.2 + Cx +
E ED x.sup.2 + Dx + B CA x.sup.2 + Ex + D FF x.sup.2 + Fx + F E4
Ground-Field Poly x.sup.4 + x.sup.3 + 1 C-F poly Basis C-F poly
Basis C-F poly Basis C-F poly Basis C-F poly Basis C-F poly Basis
C-F poly Basis C-F poly Basis x.sup.2 + 1x + 2 62 x.sup.2 + 2x + 8
32 x.sup.2 + 3x + 1 23 x.sup.2 + 4x + 2 30 x.sup.2 + 5x + 1 45
x.sup.2 + 6x + 1 19 x.sup.2 + 7x + 1 12 x.sup.2 + 8x + 1 98 x.sup.2
+ 1x + 2 64 x.sup.2 + 2x + 8 34 x.sup.2 + 3x + 1 25 x.sup.2 + 4x +
2 3C x.sup.2 + 5x + 1 48 x.sup.2 + 6x + 1 1F x.sup.2 + 7x + 1 15
x.sup.2 + 8x + 1 9F x.sup.2 + 1x + 2 72 x.sup.2 + 2x + 8 65 x.sup.2
+ 3x + 1 47 x.sup.2 + 4x + 2 55 x.sup.2 + 5x + 1 61 x.sup.2 + 6x +
1 26 x.sup.2 + 7x + 1 75 x.sup.2 + 8x + 1 A0 x.sup.2 + 1x + 2 75
x.sup.2 + 2x + 8 69 x.sup.2 + 3x + 1 4B x.sup.2 + 4x + 2 58 x.sup.2
+ 5x + 1 66 x.sup.2 + 6x + 1 2A x.sup.2 + 7x + 1 79 x.sup.2 + 8x +
1 A6 x.sup.2 + 1x + 2 C5 x.sup.2 + 2x + 8 A7 x.sup.2 + 3x + 1 A0
x.sup.2 + 4x + 2 B1 x.sup.2 + 5x + 1 9A x.sup.2 + 6x + 1 59 x.sup.2
+ 7x + 1 97 x.sup.2 + 8x + 1 D1 x.sup.2 + 1x + 2 C9 x.sup.2 + 2x +
8 AA x.sup.2 + 3x + 1 A7 x.sup.2 + 4x + 2 B6 x.sup.2 + 5x + 1 9C
x.sup.2 + 6x + 1 5E x.sup.2 + 7x + 1 9A x.sup.2 + 8x + 1 DD x.sup.2
+ 1x + 2 D7 x.sup.2 + 2x + 8 F2 x.sup.2 + 3x + 1 C1 x.sup.2 + 4x +
2 DA x.sup.2 + 5x + 1 B0 x.sup.2 + 6x + 1 62 x.sup.2 + 7x + 1 F2
x.sup.2 + 8x + 1 E6 x.sup.2 + 1x + 2 DA x.sup.2 + 2x + 8 F5 x.sup.2
+ 3x + 1 CC x.sup.2 + 4x + 2 DC x.sup.2 + 5x + 1 BC x.sup.2 + 6x +
1 6F x.sup.2 + 7x + 1 F4 x.sup.2 + 8x + 1 EB x.sup.2 + 1x + 3 6A
x.sup.2 + 2x + 9 38 x.sup.2 + 3x + 3 21 x.sup.2 + 4x + 3 32 x.sup.2
+ 5x + 3 46 x.sup.2 + 6x + 3 12 x.sup.2 + 7x + 2 1B x.sup.2 + 8x +
3 90 x.sup.2 + 1x + 3 6C x.sup.2 + 2x + 9 3E x.sup.2 + 3x + 3 27
x.sup.2 + 4x + 3 3E x.sup.2 + 5x + 3 4B x.sup.2 + 6x + 3 14 x.sup.2
+ 7x + 2 1C x.sup.2 + 8x + 3 97 x.sup.2 + 1x + 3 71 x.sup.2 + 2x +
9 64 x.sup.2 + 3x + 3 43 x.sup.2 + 4x + 3 53 x.sup.2 + 5x + 3 68
x.sup.2 + 6x + 3 25 x.sup.2 + 7x + 2 74 x.sup.2 + 8x + 3 A3 x.sup.2
+ 1x + 3 76 x.sup.2 + 2x + 9 68 x.sup.2 + 3x + 3 4F x.sup.2 + 4x +
3 5B x.sup.2 + 5x + 3 6F x.sup.2 + 6x + 3 29 x.sup.2 + 7x + 2 78
x.sup.2 + 8x + 3 A5 x.sup.2 + 1x + 3 C0 x.sup.2 + 2x + 9 A4 x.sup.2
+ 3x + 3 AA x.sup.2 + 4x + 3 B3 x.sup.2 + 5x + 3 90 x.sup.2 + 6x +
3 52 x.sup.2 + 7x + 2 94 x.sup.2 + 8x + 3 D7 x.sup.2 + 1x + 3 CC
x.sup.2 + 2x + 9 A9 x.sup.2 + 3x + 3 AD x.sup.2 + 4x + 3 B4 x.sup.2
+ 5x + 3 96 x.sup.2 + 6x + 3 55 x.sup.2 + 7x + 2 99 x.sup.2 + 8x +
3 DB x.sup.2 + 1x + 3 D5 x.sup.2 + 2x + 9 FB x.sup.2 + 3x + 3 C0
x.sup.2 + 4x + 3 DB x.sup.2 + 5x + 3 B1 x.sup.2 + 6x + 3 67 x.sup.2
+ 7x + 2 F8 x.sup.2 + 8x + 3 E1 x.sup.2 + 1x + 3 D8 x.sup.2 + 2x +
9 FC x.sup.2 + 3x + 3 CD x.sup.2 + 4x + 3 DD x.sup.2 + 5x + 3 BD
x.sup.2 + 6x + 3 6A x.sup.2 + 7x + 2 FE x.sup.2 + 8x + 3 EC x.sup.2
+ 1x + 4 68 x.sup.2 + 2x + A 3B x.sup.2 + 3x + 4 2B x.sup.2 + 4x +
6 31 x.sup.2 + 5x + 5 41 x.sup.2 + 6x + 4 13 x.sup.2 + 7x + 4 1A
x.sup.2 + 8x + 5 9A x.sup.2 + 1x + 4 6E x.sup.2 + 2x + A 3D x.sup.2
+ 3x + 4 2D x.sup.2 + 4x + 6 3D x.sup.2 + 5x + 5 4C x.sup.2 + 6x +
4 15 x.sup.2 + 7x + 4 1D x.sup.2 + 8x + 5 9D x.sup.2 + 1x + 4 7B
x.sup.2 + 2x + A 62 x.sup.2 + 3x + 4 42 x.sup.2 + 4x + 6 56 x.sup.2
+ 5x + 5 60 x.sup.2 + 6x + 4 27 x.sup.2 + 7x + 4 73 x.sup.2 + 8x +
5 A1 x.sup.2 + 1x + 4 7C x.sup.2 + 2x + A 6E x.sup.2 + 3x + 4 4E
x.sup.2 + 4x + 6 5B x.sup.2 + 5x + 5 67 x.sup.2 + 6x + 4 2B x.sup.2
+ 7x + 4 7F x.sup.2 + 8x + 5 A7 x.sup.2 + 1x + 4 C4 x.sup.2 + 2x +
A A3 x.sup.2 + 3x + 4 A3 x.sup.2 + 4x + 6 B8 x.sup.2 + 5x + 5 93
x.sup.2 + 6x + 4 50 x.sup.2 + 7x + 4 90 x.sup.2 + 8x + 5 D3 x.sup.2
+ 1x + 4 C8 x.sup.2 + 2x + A AE x.sup.2 + 3x + 4 A4 x.sup.2 + 4x +
6 BF x.sup.2 + 5x + 5 95 x.sup.2 + 6x + 4 57 x.sup.2 + 7x + 4 9D
x.sup.2 + 8x + 5 DF x.sup.2 + 1x + 4 D4 x.sup.2 + 2x + A F3 x.sup.2
+ 3x + 4 C3 x.sup.2 + 4x + 6 D0 x.sup.2 + 5x + 5 B7 x.sup.2 + 6x +
4 61 x.sup.2 + 7x + 4 F1 x.sup.2 + 8x + 5 E0 x.sup.2 + 1x + 4 D9
x.sup.2 + 2x + A F4 x.sup.2 + 3x + 4 CE x.sup.2 + 4x + 6 D6 x.sup.2
+ 5x + 5 BB x.sup.2 + 6x + 4 6C x.sup.2 + 7x + 4 F7 x.sup.2 + 8x +
5 ED x.sup.2 + 1x + 5 60 x.sup.2 + 2x + B 31 x.sup.2 + 3x + 6 29
x.sup.2 + 4x + 7 33 x.sup.2 + 5x + 7 42 x.sup.2 + 6x + 6 18 x.sup.2
+ 7x + 7 13 x.sup.2 + 8x + 7 92 x.sup.2 + 1x + 5 66 x.sup.2 + 2x +
B 37 x.sup.2 + 3x + 6 2F x.sup.2 + 4x + 7 3F x.sup.2 + 5x + 7 4F
x.sup.2 + 6x + 6 1E x.sup.2 + 7x + 7 14 x.sup.2 + 8x + 7 95 x.sup.2
+ 1x + 5 78 x.sup.2 + 2x + B 63 x.sup.2 + 3x + 6 46 x.sup.2 + 4x +
7 50 x.sup.2 + 5x + 7 69 x.sup.2 + 6x + 6 24 x.sup.2 + 7x + 7 72
x.sup.2 + 8x + 7 A2 x.sup.2 + 1x + 5 7F x.sup.2 + 2x + B 6F x.sup.2
+ 3x + 6 4A x.sup.2 + 4x + 7 5D x.sup.2 + 5x + 7 6E x.sup.2 + 6x +
6 28 x.sup.2 + 7x + 7 7E x.sup.2 + 8x + 7 A4 x.sup.2 + 1x + 5 C1
x.sup.2 + 2x + B A0 x.sup.2 + 3x + 6 A9 x.sup.2 + 4x + 7 BA x.sup.2
+ 5x + 7 99 x.sup.2 + 6x + 6 5B x.sup.2 + 7x + 7 93 x.sup.2 + 8x +
7 D5 x.sup.2 + 1x + 5 CD x.sup.2 + 2x + B AD x.sup.2 + 3x + 6 AE
x.sup.2 + 4x + 7 BD x.sup.2 + 5x + 7 9F x.sup.2 + 6x + 6 5C x.sup.2
+ 7x + 7 9E x.sup.2 + 8x + 7 D9 x.sup.2 + 1x + 5 D6 x.sup.2 + 2x +
B FA x.sup.2 + 3x + 6 C2 x.sup.2 + 4x + 7 D1 x.sup.2 + 5x + 7 B6
x.sup.2 + 6x + 6 64 x.sup.2 + 7x + 7 FB x.sup.2 + 8x + 7 E7 x.sup.2
+ 1x + 5 DB x.sup.2 + 2x + B FD x.sup.2 + 3x + 6 CF x.sup.2 + 4x +
7 D7 x.sup.2 + 5x + 7 8A x.sup.2 + 6x + 6 69 x.sup.2 + 7x + 7 FD
x.sup.2 + 8x + 7 EA x.sup.2 + 1x + 8 61 x.sup.2 + 2x + C 3A x.sup.2
+ 3x + 8 20 x.sup.2 + 4x + A 37 x.sup.2 + 5x + 9 47 x.sup.2 + 6x +
9 1B x.sup.2 + 7x + 8 19 x.sup.2 + 8x + 8 91 x.sup.2 + 1x + 8 67
x.sup.2 + 2x + C 3C x.sup.2 + 3x + 8 26 x.sup.2 + 4x + A 3B x.sup.2
+ 5x + 9 4A x.sup.2 + 6x + 9 1D x.sup.2 + 7x + 8 1E x.sup.2 + 8x +
8 96 x.sup.2 + 1x + 8 7A x.sup.2 + 2x + C 60 x.sup.2 + 3x + 8 41
x.sup.2 + 4x + A 51 x.sup.2 + 5x + 9 62 x.sup.2 + 6x + 9 22 x.sup.2
+ 7x + 8 76 x.sup.2 + 8x + 8 AA x.sup.2 + 1x + 8 7D x.sup.2 + 2x +
C 6C x.sup.2 + 3x + 8 4D x.sup.2 + 4x + A 5C x.sup.2 + 5x + 9 65
x.sup.2 + 6x + 9 2E x.sup.2 + 7x + 8 7A x.sup.2 + 8x + 8 AC x.sup.2
+ 1x + 8 C3 x.sup.2 + 2x + C A5 x.sup.2 + 3x + 8 A8 x.sup.2 + 4x +
A B0 x.sup.2 + 5x + 9 92 x.sup.2 + 6x + 9 53 x.sup.2 + 7x + 8 92
x.sup.2 + 8x + 8 D0 x.sup.2 + 1x + 8 CF x.sup.2 + 2x + C A8 x.sup.2
+ 3x + 8 AF x.sup.2 + 4x + A B7 x.sup.2 + 5x + 9 94 x.sup.2 + 6x +
9 54 x.sup.2 + 7x + 8 9F x.sup.2 + 8x + 8 DC x.sup.2 + 1x + 8 D0
x.sup.2 + 2x + C F1 x.sup.2 + 3x + 8 C6 x.sup.2 + 4x + A D3 x.sup.2
+ 5x + 9 B5 x.sup.2 + 6x + 9 63 x.sup.2 + 7x + 8 F9 x.sup.2 + 8x +
8 E5 x.sup.2 + 1x + 8 DD x.sup.2 + 2x + C F6 x.sup.2 + 3x + 8 CB
x.sup.2 + 4x + A D5 x.sup.2 + 5x + 9 B9 x.sup.2 + 6x + 9 6E x.sup.2
+ 7x + 8 FF x.sup.2 + 8x + 8 E8 x.sup.2 + 1x + 9 69 x.sup.2 + 2x +
D 30 x.sup.2 + 3x + A 22 x.sup.2 + 4x + B 35 x.sup.2 + 5x + B 44
x.sup.2 + 6x + B 10 x.sup.2 + 7x + B 10 x.sup.2 + 8x + A 99 x.sup.2
+ 1x + 9 6F x.sup.2 + 2x + D 36 x.sup.2 + 3x + A 24 x.sup.2 + 4x +
B 39 x.sup.2 + 5x + B 49 x.sup.2 + 6x + B 16 x.sup.2 + 7x + B 17
x.sup.2 + 8x + A 9E x.sup.2 + 1x + 9 79 x.sup.2 + 2x + D 61 x.sup.2
+ 3x + A 45 x.sup.2 + 4x + B 57 x.sup.2 + 5x + B 6B x.sup.2 + 6x +
B 21 x.sup.2 + 7x + B 77 x.sup.2 + 8x + A A9 x.sup.2 + 1x + 9 7E
x.sup.2 + 2x + D 6D x.sup.2 + 3x + A 49 x.sup.2 + 4x + B 5A x.sup.2
+ 5x + B 6C x.sup.2 + 6x + B 2D x.sup.2 + 7x + B 7B x.sup.2 + 8x +
A AF x.sup.2 + 1x + 9 C6 x.sup.2 + 2x + D A6 x.sup.2 + 3x + A A2
x.sup.2 + 4x + B B2 x.sup.2 + 5x + B 98 x.sup.2 + 6x + B 58 x.sup.2
+ 7x + B 91 x.sup.2 + 8x + A D6 x.sup.2 + 1x + 9 CA x.sup.2 + 2x +
D AB x.sup.2 + 3x + A A5 x.sup.2 + 4x + B B5 x.sup.2 + 5x + B 9E
x.sup.2 + 6x + B 5F x.sup.2 + 7x + B 9C x.sup.2 + 8x + A DA x.sup.2
+ 1x + 9 D2 x.sup.2 + 2x + D F8 x.sup.2 + 3x + A C7 x.sup.2 + 4x +
B D2 x.sup.2 + 5x + B B4 x.sup.2 + 6x + B 66 x.sup.2 + 7x + B F3
x.sup.2 + 8x + A E2 x.sup.2 + 1x + 9 DF x.sup.2 + 2x + D FF x.sup.2
+ 3x + A CA x.sup.2 + 4x + B D4 x.sup.2 + 5x + B B8 x.sup.2 + 6x +
B 6B x.sup.2 + 7x + B F5 x.sup.2 + 8x + A EF x.sup.2 + 1x + E 6B
x.sup.2 + 2x + E 33 x.sup.2 + 3x + D 28 x.sup.2 + 4x + E 36 x.sup.2
+ 5x + D 43 x.sup.2 + 6x + C 11 x.sup.2 + 7x + D 11 x.sup.2 + 8x +
C 93 x.sup.2 + 1x + E 6D x.sup.2 + 2x + E 35 x.sup.2 + 3x + D 2E
x.sup.2 + 4x + E 3A x.sup.2 + 5x + D 4E x.sup.2 + 6x + C 17 x.sup.2
+ 7x + D 16 x.sup.2 + 8x + C 94 x.sup.2 + 1x + E 73 x.sup.2 + 2x +
E 67 x.sup.2 + 3x + D 44 x.sup.2 + 4x + E 52 x.sup.2 + 5x + D 63
x.sup.2 + 6x + C 23 x.sup.2 + 7x + D 70 x.sup.2 + 8x + C AB x.sup.2
+ 1x + E 74 x.sup.2 + 2x + E 6B x.sup.2 + 3x + D 48 x.sup.2 + 4x +
E 5F x.sup.2 + 5x + D 64 x.sup.2 + 6x + C 2F x.sup.2 + 7x + D 7C
x.sup.2 + 8x + C AD x.sup.2 + 1x + E C2 x.sup.2 + 2x + E A1 x.sup.2
+ 3x + D AB x.sup.2 + 4x + E B9 x.sup.2 + 5x + D 9B x.sup.2 + 6x +
C 5A x.sup.2 + 7x + D 95 x.sup.2 + 8x + C D2 x.sup.2 + 1x + E CE
x.sup.2 + 2x + E AC x.sup.2 + 3x + D AC x.sup.2 + 4x + E BE x.sup.2
+ 5x + D 9D x.sup.2 + 6x + C 5D x.sup.2 + 7x + D 98 x.sup.2 + 8x +
C DE x.sup.2 + 1x + E D3 x.sup.2 + 2x + E F0 x.sup.2 + 3x + D C4
x.sup.2 + 4x + E D9 x.sup.2 + 5x + D B2 x.sup.2 + 6x + C 60 x.sup.2
+ 7x + D FA x.sup.2 + 8x + C E3 x.sup.2 + 1x + E DE x.sup.2 + 2x +
E F7 x.sup.2 + 3x + D C9 x.sup.2 + 4x +
E DF x.sup.2 + 5x + D BE x.sup.2 + 6x + C 6D x.sup.2 + 7x + D FC
x.sup.2 + 8x + C EE x.sup.2 + 1x + F 63 x.sup.2 + 2x + F 39 x.sup.2
+ 3x + F 2A x.sup.2 + 4x + F 34 x.sup.2 + 5x + F 40 x.sup.2 + 6x +
E 1A x.sup.2 + 7x + E 18 x.sup.2 + 8x + E 9B x.sup.2 + 1x + F 65
x.sup.2 + 2x + F 3F x.sup.2 + 3x + F 2C x.sup.2 + 4x + F 38 x.sup.2
+ 5x + F 4D x.sup.2 + 6x + E 1C x.sup.2 + 7x + E 1F x.sup.2 + 8x +
E 9C x.sup.2 + 1x + F 70 x.sup.2 + 2x + F 66 x.sup.2 + 3x + F 40
x.sup.2 + 4x + F 54 x.sup.2 + 5x + F 6A x.sup.2 + 6x + E 20 x.sup.2
+ 7x + E 71 x.sup.2 + 8x + E A8 x.sup.2 + 1x + F 77 x.sup.2 + 2x +
F 6A x.sup.2 + 3x + F 4C x.sup.2 + 4x + F 59 x.sup.2 + 5x + F 6D
x.sup.2 + 6x + E 2C x.sup.2 + 7x + E 7D x.sup.2 + 8x + E AE x.sup.2
+ 1x + F C7 x.sup.2 + 2x + F A2 x.sup.2 + 3x + F A1 x.sup.2 + 4x +
F BB x.sup.2 + 5x + F 91 x.sup.2 + 6x + E 51 x.sup.2 + 7x + E 96
x.sup.2 + 8x + E D4 x.sup.2 + 1x + F CB x.sup.2 + 2x + F AF x.sup.2
+ 3x + F A6 x.sup.2 + 4x + F BC x.sup.2 + 5x + F 97 x.sup.2 + 6x +
E 56 x.sup.2 + 7x + E 9B x.sup.2 + 8x + E D8 x.sup.2 + 1x + F D1
x.sup.2 + 2x + F F9 x.sup.2 + 3x + F C5 x.sup.2 + 4x + F D8 x.sup.2
+ 5x + F B3 x.sup.2 + 6x + E 65 x.sup.2 + 7x + E F0 x.sup.2 + 8x +
E E4 x.sup.2 + 1x + F DC x.sup.2 + 2x + F FE x.sup.2 + 3x + F C8
x.sup.2 + 4x + F DE x.sup.2 + 5x + F BF x.sup.2 + 6x + E 68 x.sup.2
+ 7x + E F6 x.sup.2 + 8x + E E9 Ground-Field Poly x.sup.4 + x.sup.3
+ 1 C-F poly Basis C-F poly Basis C-F poly Basis C-F poly Basis C-F
poly Basis C-F poly Basis C-F poly Basis x.sup.2 + 9x + 2 5B
x.sup.2 + Ax + 4 25 x.sup.2 + Bx + 2 40 x.sup.2 + Cx + 1 14 x.sup.2
+ Dx + 1 13 x.sup.2 + Ex + 4 87 x.sup.2 + Fx + 1 2A x.sup.2 + 9x +
2 5D x.sup.2 + Ax + 4 28 x.sup.2 + Bx + 2 47 x.sup.2 + Cx + 1 18
x.sup.2 + Dx + 1 1E x.sup.2 + Ex + 4 8A x.sup.2 + Fx + 1 2D x.sup.2
+ 9x + 2 73 x.sup.2 + Ax + 4 31 x.sup.2 + Bx + 2 57 x.sup.2 + Cx +
1 34 x.sup.2 + Dx + 1 4B x.sup.2 + Ex + 4 B2 x.sup.2 + Fx + 1 71
x.sup.2 + 9x + 2 7E x.sup.2 + Ax + 4 36 x.sup.2 + Bx + 2 5B x.sup.2
+ Cx + 1 39 x.sup.2 + Dx + 1 4D x.sup.2 + Ex + 4 B4 x.sup.2 + Fx +
1 77 x.sup.2 + 9x + 2 83 x.sup.2 + Ax + 4 8A x.sup.2 + Bx + 2 E3
x.sup.2 + Cx + 1 C8 x.sup.2 + Dx + 1 82 x.sup.2 + Ex + 4 C2 x.sup.2
+ Fx + 1 B0 x.sup.2 + 9x + 2 84 x.sup.2 + Ax + 4 8C x.sup.2 + Bx +
2 E5 x.sup.2 + Cx + 1 CE x.sup.2 + Dx + 1 8E x.sup.2 + Ex + 4 C5
x.sup.2 + Fx + 1 BD x.sup.2 + 9x + 2 A2 x.sup.2 + Ax + 4 90 x.sup.2
+ Bx + 2 F1 x.sup.2 + Cx + 1 EB x.sup.2 + Dx + 1 D3 x.sup.2 + Ex +
4 F5 x.sup.2 + Fx + 1 E3 x.sup.2 + 9x + 2 AE x.sup.2 + Ax + 4 9C
x.sup.2 + Bx + 2 FC x.sup.2 + Cx + 1 EC x.sup.2 + Dx + 1 D4 x.sup.2
+ Ex + 4 F9 x.sup.2 + Fx + 1 EF x.sup.2 + 9x + 3 59 x.sup.2 + Ax +
5 22 x.sup.2 + Bx + 3 4B x.sup.2 + Cx + 2 13 x.sup.2 + Dx + 2 16
x.sup.2 + Ex + 5 83 x.sup.2 + Fx + 2 29 x.sup.2 + 9x + 3 5F x.sup.2
+ Ax + 5 2F x.sup.2 + Bx + 3 4C x.sup.2 + Cx + 2 1F x.sup.2 + Dx +
2 1B x.sup.2 + Ex + 5 8E x.sup.2 + Fx + 2 2E x.sup.2 + 9x + 3 72
x.sup.2 + Ax + 5 39 x.sup.2 + Bx + 3 54 x.sup.2 + Cx + 2 30 x.sup.2
+ Dx + 2 40 x.sup.2 + Ex + 5 BB x.sup.2 + Fx + 2 79 x.sup.2 + 9x +
3 7F x.sup.2 + Ax + 5 3E x.sup.2 + Bx + 3 58 x.sup.2 + Cx + 2 3D
x.sup.2 + Dx + 2 46 x.sup.2 + Ex + 5 BD x.sup.2 + Fx + 2 7F x.sup.2
+ 9x + 3 89 x.sup.2 + Ax + 5 89 x.sup.2 + Bx + 3 E8 x.sup.2 + Cx +
2 C1 x.sup.2 + Dx + 2 81 x.sup.2 + Ex + 5 C3 x.sup.2 + Fx + 2 B2
x.sup.2 + 9x + 3 8E x.sup.2 + Ax + 5 8F x.sup.2 + Bx + 3 EE x.sup.2
+ Cx + 2 C7 x.sup.2 + Dx + 2 8D x.sup.2 + Ex + 5 C4 x.sup.2 + Fx +
2 BF x.sup.2 + 9x + 3 A6 x.sup.2 + Ax + 5 96 x.sup.2 + Bx + 3 F4
x.sup.2 + Cx + 2 EA x.sup.2 + Dx + 2 D8 x.sup.2 + Ex + 5 F2 x.sup.2
+ Fx + 2 E6 x.sup.2 + 9x + 3 AA x.sup.2 + Ax + 5 9A x.sup.2 + Bx +
3 F9 x.sup.2 + Cx + 2 ED x.sup.2 + Dx + 2 DF x.sup.2 + Ex + 5 FE
x.sup.2 + Fx + 2 EA x.sup.2 + 9x + 4 50 x.sup.2 + Ax + 6 23 x.sup.2
+ Bx + 6 4A x.sup.2 + Cx + 4 16 x.sup.2 + Dx + 5 14 x.sup.2 + Ex +
6 85 x.sup.2 + Fx + 5 21 x.sup.2 + 9x + 4 56 x.sup.2 + Ax + 6 2E
x.sup.2 + Bx + 6 4D x.sup.2 + Cx + 4 1A x.sup.2 + Dx + 5 19 x.sup.2
+ Ex + 6 88 x.sup.2 + Fx + 5 26 x.sup.2 + 9x + 4 76 x.sup.2 + Ax +
6 33 x.sup.2 + Bx + 6 53 x.sup.2 + Cx + 4 32 x.sup.2 + Dx + 5 48
x.sup.2 + Ex + 6 BA x.sup.2 + Fx + 5 7A x.sup.2 + 9x + 4 7B x.sup.2
+ Ax + 6 34 x.sup.2 + Bx + 6 5F x.sup.2 + Cx + 4 3F x.sup.2 + Dx +
5 4E x.sup.2 + Ex + 6 BC x.sup.2 + Fx + 5 7C x.sup.2 + 9x + 4 88
x.sup.2 + Ax + 6 8B x.sup.2 + Bx + 6 E1 x.sup.2 + Cx + 4 C9 x.sup.2
+ Dx + 5 84 x.sup.2 + Ex + 6 C1 x.sup.2 + Fx + 5 B5 x.sup.2 + 9x +
4 8F x.sup.2 + Ax + 6 8D x.sup.2 + Bx + 6 E7 x.sup.2 + Cx + 4 CF
x.sup.2 + Dx + 5 88 x.sup.2 + Ex + 6 C6 x.sup.2 + Fx + 5 B8 x.sup.2
+ 9x + 4 A1 x.sup.2 + Ax + 6 92 x.sup.2 + Bx + 6 F0 x.sup.2 + Cx +
4 E9 x.sup.2 + Dx + 5 DB x.sup.2 + Ex + 6 F0 x.sup.2 + Fx + 5 E0
x.sup.2 + 9x + 4 AD x.sup.2 + Ax + 6 9E x.sup.2 + Bx + 6 FD x.sup.2
+ Cx + 4 EE x.sup.2 + Dx + 5 DC x.sup.2 + Ex + 6 FC x.sup.2 + Fx +
5 EC x.sup.2 + 9x + 5 52 x.sup.2 + Ax + 7 24 x.sup.2 + Bx + 7 41
x.sup.2 + Cx + 7 11 x.sup.2 + Dx + 6 11 x.sup.2 + Ex + 7 81 x.sup.2
+ Fx + 6 22 x.sup.2 + 9x + 5 54 x.sup.2 + Ax + 7 29 x.sup.2 + Bx +
7 46 x.sup.2 + Cx + 7 1D x.sup.2 + Dx + 6 1C x.sup.2 + Ex + 7 8C
x.sup.2 + Fx + 6 25 x.sup.2 + 9x + 5 77 x.sup.2 + Ax + 7 3B x.sup.2
+ Bx + 7 50 x.sup.2 + Cx + 7 36 x.sup.2 + Dx + 6 43 x.sup.2 + Ex +
7 B3 x.sup.2 + Fx + 6 72 x.sup.2 + 9x + 5 7A x.sup.2 + Ax + 7 3C
x.sup.2 + Bx + 7 5C x.sup.2 + Cx + 7 3B x.sup.2 + Dx + 6 45 x.sup.2
+ Ex + 7 B5 x.sup.2 + Fx + 6 74 x.sup.2 + 9x + 5 82 x.sup.2 + Ax +
7 88 x.sup.2 + Bx + 7 EA x.sup.2 + Cx + 7 C0 x.sup.2 + Dx + 6 87
x.sup.2 + Ex + 7 C0 x.sup.2 + Fx + 6 B7 x.sup.2 + 9x + 5 85 x.sup.2
+ Ax + 7 8E x.sup.2 + Bx + 7 EC x.sup.2 + Cx + 7 C6 x.sup.2 + Dx +
6 8B x.sup.2 + Ex + 7 C7 x.sup.2 + Fx + 6 BA x.sup.2 + 9x + 5 A5
x.sup.2 + Ax + 7 94 x.sup.2 + Bx + 7 F5 x.sup.2 + Cx + 7 E8 x.sup.2
+ Dx + 6 D0 x.sup.2 + Ex + 7 F7 x.sup.2 + Fx + 6 E5 x.sup.2 + 9x +
5 A9 x.sup.2 + Ax + 7 98 x.sup.2 + Bx + 7 F8 x.sup.2 + Cx + 7 EF
x.sup.2 + Dx + 6 D7 x.sup.2 + Ex + 7 FB x.sup.2 + Fx + 6 E9 x.sup.2
+ 9x + A 58 x.sup.2 + Ax + C 26 x.sup.2 + Bx + 8 43 x.sup.2 + Cx +
9 17 x.sup.2 + Dx + 9 15 x.sup.2 + Ex + 8 84 x.sup.2 + Fx + 8 20
x.sup.2 + 9x + A 5E x.sup.2 + Ax + C 2B x.sup.2 + Bx + 8 44 x.sup.2
+ Cx + 9 1B x.sup.2 + Dx + 9 18 x.sup.2 + Ex + 8 89 x.sup.2 + Fx +
8 27 x.sup.2 + 9x + A 74 x.sup.2 + Ax + C 38 x.sup.2 + Bx + 8 52
x.sup.2 + Cx + 9 31 x.sup.2 + Dx + 9 4A x.sup.2 + Ex + 8 B8 x.sup.2
+ Fx + 8 73 x.sup.2 + 9x + A 79 x.sup.2 + Ax + C 3F x.sup.2 + Bx +
8 5E x.sup.2 + Cx + 9 3C x.sup.2 + Dx + 9 4C x.sup.2 + Ex + 8 BE
x.sup.2 + Fx + 8 75 x.sup.2 + 9x + A 8B x.sup.2 + Ax + C 80 x.sup.2
+ Bx + 8 EB x.sup.2 + Cx + 9 C3 x.sup.2 + Dx + 9 80 x.sup.2 + Ex +
8 CB x.sup.2 + Fx + 8 B1 x.sup.2 + 9x + A 8C x.sup.2 + Ax + C 86
x.sup.2 + Bx + 8 ED x.sup.2 + Cx + 9 C5 x.sup.2 + Dx + 9 8C x.sup.2
+ Ex + 8 CC x.sup.2 + Fx + 8 BC x.sup.2 + 9x + A A4 x.sup.2 + Ax +
C 91 x.sup.2 + Bx + 8 F3 x.sup.2 + Cx + 9 E0 x.sup.2 + Dx + 9 D1
x.sup.2 + Ex + 8 F4 x.sup.2 + Fx + 8 E7 x.sup.2 + 9x + A A8 x.sup.2
+ Ax + C 9D x.sup.2 + Bx + 8 FE x.sup.2 + Cx + 9 E7 x.sup.2 + Dx +
9 D6 x.sup.2 + Ex + 8 F8 x.sup.2 + Fx + 8 EB x.sup.2 + 9x + B 5A
x.sup.2 + Ax + D 21 x.sup.2 + Bx + 9 48 x.sup.2 + Cx + A 10 x.sup.2
+ Dx + A 10 x.sup.2 + Ex + 9 80 x.sup.2 + Fx + B 23 x.sup.2 + 9x +
B 5C x.sup.2 + Ax + D 2C x.sup.2 + Bx + 9 4F x.sup.2 + Cx + A 1C
x.sup.2 + Dx + A 1D x.sup.2 + Ex + 9 8D x.sup.2 + Fx + B 24 x.sup.2
+ 9x + B 75 x.sup.2 + Ax + D 30 x.sup.2 + Bx + 9 51 x.sup.2 + Cx +
A 35 x.sup.2 + Dx + A 41 x.sup.2 + Ex + 9 B1 x.sup.2 + Fx + B 7B
x.sup.2 + 9x + B 78 x.sup.2 + Ax + D 37 x.sup.2 + Bx + 9 5D x.sup.2
+ Cx + A 38 x.sup.2 + Dx + A 47 x.sup.2 + Ex + 9 B7 x.sup.2 + Fx +
B 7D x.sup.2 + 9x + B 81 x.sup.2 + Ax + D 83 x.sup.2 + Bx + 9 E0
x.sup.2 + Cx + A CA x.sup.2 + Dx + A 83 x.sup.2 + Ex + 9 CA x.sup.2
+ Fx + B B3 x.sup.2 + 9x + B 86 x.sup.2 + Ax + D 85 x.sup.2 + Bx +
9 E6 x.sup.2 + Cx + A CC x.sup.2 + Dx + A 8F x.sup.2 + Ex + 9 CD
x.sup.2 + Fx + B BE x.sup.2 + 9x + B A0 x.sup.2 + Ax + D 97 x.sup.2
+ Bx + 9 F6 x.sup.2 + Cx + A E1 x.sup.2 + Dx + A DA x.sup.2 + Ex +
9 F3 x.sup.2 + Fx + B E2 x.sup.2 + 9x + B AC x.sup.2 + Ax + D 9B
x.sup.2 + Bx + 9 FB x.sup.2 + Cx + A E6 x.sup.2 + Dx + A DD x.sup.2
+ Ex + 9 FF x.sup.2 + Fx + B EE x.sup.2 + 9x + C 53 x.sup.2 + Ax +
E 20 x.sup.2 + Bx + C 49 x.sup.2 + Cx + C 15 x.sup.2 + Dx + D 12
x.sup.2 + Ex + A 86 x.sup.2 + Fx + C 2B x.sup.2 + 9x + C 55 x.sup.2
+ Ax + E 2D x.sup.2 + Bx + C 4E x.sup.2 + Cx + C 19 x.sup.2 + Dx +
D 1F x.sup.2 + Ex + A 8B x.sup.2 + Fx + C 2C x.sup.2 + 9x + C 71
x.sup.2 + Ax + E 3A x.sup.2 + Bx + C 56 x.sup.2 + Cx + C 37 x.sup.2
+ Dx + D 49 x.sup.2 + Ex + A B0 x.sup.2 + Fx + C 78 x.sup.2 + 9x +
C 7C x.sup.2 + Ax + E 3D x.sup.2 + Bx + C 5A x.sup.2 + Cx + C 3A
x.sup.2 + Dx + D 4F x.sup.2 + Ex + A B6 x.sup.2 + Fx + C 7E x.sup.2
+ 9x + C 80 x.sup.2 + Ax + E 81 x.sup.2 + Bx + C E9 x.sup.2 + Cx +
C C2 x.sup.2 + Dx + D 86 x.sup.2 + Ex + A C8 x.sup.2 + Fx + C B4
x.sup.2 + 9x + C 87 x.sup.2 + Ax + E 87 x.sup.2 + Bx + C EF x.sup.2
+ Cx + C C4 x.sup.2 + Dx + D 8A x.sup.2 + Ex + A CF x.sup.2 + Fx +
C B9 x.sup.2 + 9x + C A7 x.sup.2 + Ax + E 93 x.sup.2 + Bx + C F2
x.sup.2 + Cx + C E2 x.sup.2 + Dx + D D9 x.sup.2 + Ex + A F1 x.sup.2
+ Fx + C E4 x.sup.2 + 9x + C AB x.sup.2 + Ax + E 9F x.sup.2 + Bx +
C FF x.sup.2 + Cx + C E5 x.sup.2 + Dx + D DE x.sup.2 + Ex + A FD
x.sup.2 + Fx + C E8 x.sup.2 + 9x + D 51 x.sup.2 + Ax + F 27 x.sup.2
+ Bx + D 42 x.sup.2 + Cx + F 12 x.sup.2 + Dx + E 17 x.sup.2 + Ex +
B 82 x.sup.2 + Fx + F 28 x.sup.2 + 9x + D 57 x.sup.2 + Ax + F 2A
x.sup.2 + Bx + D 45 x.sup.2 + Cx + F 1E x.sup.2 + Dx + E 1A x.sup.2
+ Ex + B 8F x.sup.2 + Fx + F 2F x.sup.2 + 9x + D 70 x.sup.2 + Ax +
F 32 x.sup.2 + Bx + D 55 x.sup.2 + Cx + F 33 x.sup.2 + Dx + E 42
x.sup.2 + Ex + B B9 x.sup.2 + Fx + F 70 x.sup.2 + 9x + D 7D x.sup.2
+ Ax + F 35 x.sup.2 + Bx + D 59 x.sup.2 + Cx + F 3E x.sup.2 + Dx +
E 44 x.sup.2 + Ex + B BF x.sup.2 + Fx + F 76 x.sup.2 + 9x + D 8A
x.sup.2 + Ax + F 82 x.sup.2 + Bx + D E2 x.sup.2 + Cx + F CB x.sup.2
+ Dx + E 85 x.sup.2 + Ex + B C9 x.sup.2 + Fx + F B6 x.sup.2 + 9x +
D 8D x.sup.2 + Ax + F 84 x.sup.2 + Bx + D E4 x.sup.2 + Cx + F CD
x.sup.2 + Dx + E 89 x.sup.2 + Ex + B CE x.sup.2 + Fx + F BB x.sup.2
+ 9x + D A3 x.sup.2 + Ax + F 95 x.sup.2 + Bx + D F7 x.sup.2 + Cx +
F E3 x.sup.2 + Dx + E D2 x.sup.2 + Ex + B F6 x.sup.2 + Fx + F E1
x.sup.2 + 9x + D AF x.sup.2 + Ax + F 99 x.sup.2 + Bx + D FA x.sup.2
+ Cx + F E4 x.sup.2 + Dx + E D5 x.sup.2 + Ex + B FA x.sup.2 + Fx +
F ED Ground-Field Poly = x.sup.4 + x.sup.3 + x.sup.2 + x + 1 C-F
poly Basis C-F poly Basis C-F poly Basis C-F poly Basis C-F poly
Basis C-F poly Basis C-F poly Basis C-F poly Basis x.sup.2 + 1x + 2
6A x.sup.2 + 2x + 1 32 x.sup.2 + 3x + 8 28 x.sup.2 + 4x + 1 54
x.sup.2 + 5x + 2 23 x.sup.2 + 6x + 1 18 x.sup.2 + 7x + 1 13 x.sup.2
+ 8x + 1 3B x.sup.2 + 1x + 2 6C x.sup.2 + 2x + 1 34 x.sup.2 + 3x +
8 2E x.sup.2 + 4x + 1 5F x.sup.2 + 5x + 2 29 x.sup.2 + 6x + 1 1E
x.sup.2 + 7x + 1 14 x.sup.2 + 8x + 1 3C x.sup.2 + 1x + 2 71 x.sup.2
+ 2x + 1 57 x.sup.2 + 3x + 8 66 x.sup.2 + 4x + 1 61 x.sup.2 + 5x +
2 42 x.sup.2 + 6x + 1 36 x.sup.2 + 7x + 1 74 x.sup.2 + 8x + 1 71
x.sup.2 + 1x + 2 76 x.sup.2 + 2x + 1 5D x.sup.2 + 3x + 8 6C x.sup.2
+ 4x + 1 66 x.sup.2 + 5x + 2 49 x.sup.2 + 6x + 1 3C x.sup.2 + 7x +
1 7E x.sup.2 + 8x + 1 77 x.sup.2 + 1x + 2 A0 x.sup.2 + 2x + 1 A1
x.sup.2 + 3x + 8 89 x.sup.2 + 4x + 1 D0 x.sup.2 + 5x + 2 BB x.sup.2
+ 6x + 1 49 x.sup.2 + 7x + 1 83 x.sup.2 + 8x + 1 92 x.sup.2 + 1x +
2 AA x.sup.2 + 2x + 1 AA x.sup.2 + 3x + 8 8E x.sup.2 + 4x + 1 DA
x.sup.2 + 5x + 2 BD x.sup.2 + 6x + 1 4E x.sup.2 + 7x + 1 85 x.sup.2
+ 8x + 1 98 x.sup.2 + 1x + 2 B4 x.sup.2 + 2x + 1 C0 x.sup.2 + 3x +
8 C3 x.sup.2 + 4x + 1 EA x.sup.2 + 5x + 2 D2 x.sup.2 + 6x + 1 63
x.sup.2 + 7x + 1 E7 x.sup.2 + 8x + 1 D0 x.sup.2 + 1x + 2 BF x.sup.2
+ 2x + 1 C7 x.sup.2 + 3x + 8 C8 x.sup.2 + 4x + 1 EC x.sup.2 + 5x +
2 D5 x.sup.2 + 6x + 1 68 x.sup.2 + 7x + 1 EC x.sup.2 + 8x + 1 DB
x.sup.2 + 1x + 3 63 x.sup.2 + 2x + 2 31 x.sup.2 + 3x + 9 22 x.sup.2
+ 4x + 2 56 x.sup.2 + 5x + 3 20 x.sup.2 + 6x + 2 13 x.sup.2 + 7x +
3 1A x.sup.2 + 8x + 3 39 x.sup.2 + 1x + 3 65 x.sup.2 + 2x + 2 37
x.sup.2 + 3x + 9 24 x.sup.2 + 4x + 2 5D x.sup.2 + 5x + 3 2A x.sup.2
+ 6x + 2 15 x.sup.2 + 7x + 3 1D x.sup.2 + 8x + 3 3E x.sup.2 + 1x +
3 73 x.sup.2 + 2x + 2 52 x.sup.2 + 3x + 9 67 x.sup.2 + 4x + 2 68
x.sup.2 + 5x + 3 44 x.sup.2 + 6x + 2 34 x.sup.2 + 7x + 3 75 x.sup.2
+ 8x + 3 78 x.sup.2 + 1x + 3 74 x.sup.2 + 2x + 2 58 x.sup.2 + 3x +
9 6D x.sup.2 + 4x + 2 6F x.sup.2 + 5x + 3 4F x.sup.2 + 6x + 2 3E
x.sup.2 + 7x + 3 7F x.sup.2 + 8x + 3 7E x.sup.2 + 1x + 3 A4 x.sup.2
+ 2x + 2 A0 x.sup.2 + 3x + 9 80 x.sup.2 + 4x + 2 D1 x.sup.2 + 5x +
3 BA x.sup.2 + 6x + 2 43 x.sup.2 + 7x + 3 89 x.sup.2 + 8x + 3 96
x.sup.2 + 1x + 3 AE x.sup.2 + 2x + 2 AB x.sup.2 + 3x + 9 87 x.sup.2
+ 4x + 2 DB x.sup.2 + 5x + 3 BC x.sup.2 + 6x + 2 44 x.sup.2 + 7x +
3 8F x.sup.2 + 8x + 3 9C x.sup.2 + 1x + 3 B7 x.sup.2 + 2x + 2 CB
x.sup.2 + 3x + 9 C1 x.sup.2 + 4x + 2 E0 x.sup.2 + 5x + 3 D1 x.sup.2
+ 6x + 2 67 x.sup.2 + 7x + 3 E5 x.sup.2 + 8x + 3 D3 x.sup.2 + 1x +
3 BC x.sup.2 + 2x + 2 CC x.sup.2 + 3x + 9 CA x.sup.2 + 4x + 2 E6
x.sup.2 + 5x + 3 D6 x.sup.2 + 6x + 2 6C x.sup.2 + 7x + 3 EE x.sup.2
+ 8x + 3 D8 x.sup.2 + 1x + 4 60 x.sup.2 + 2x + 5 3B x.sup.2 + 3x +
A 2A x.sup.2 + 4x + 4 51 x.sup.2 + 5x + 6 21 x.sup.2 + 6x + 5 12
x.sup.2 + 7x + 5 1B x.sup.2 + 8x + 4 31 x.sup.2 + 1x + 4 66 x.sup.2
+ 2x + 5 3D x.sup.2 + 3x + A 2C x.sup.2 + 4x + 4 5A x.sup.2 + 5x +
6 2B x.sup.2 + 6x + 5 14 x.sup.2 + 7x + 5 1C x.sup.2 + 8x + 4 36
x.sup.2 + 1x + 4 78 x.sup.2 + 2x + 5 53 x.sup.2 + 3x + A 60 x.sup.2
+ 4x + 4 60 x.sup.2 + 5x + 6 46 x.sup.2 + 6x + 5 37 x.sup.2 + 7x +
5 72 x.sup.2 + 8x + 4 7A x.sup.2 + 1x + 4 7F x.sup.2 + 2x + 5 59
x.sup.2 + 3x + A 6A x.sup.2 + 4x + 4 67 x.sup.2 + 5x + 6 4D x.sup.2
+ 6x + 5 3D x.sup.2 + 7x + 5 78 x.sup.2 + 8x + 4 7C x.sup.2 + 1x +
4 A1 x.sup.2 + 2x + 5 A2 x.sup.2 + 3x + A 81 x.sup.2 + 4x + 4 D7
x.sup.2 + 5x + 6 B0 x.sup.2 + 6x + 5 40 x.sup.2 + 7x + 5 81 x.sup.2
+ 8x + 4 90 x.sup.2 + 1x + 4 AB x.sup.2 + 2x + 5 A9 x.sup.2 + 3x +
A 86 x.sup.2 + 4x + 4 DD x.sup.2 + 5x + 6 B6 x.sup.2 + 6x + 5 47
x.sup.2 + 7x + 5 87 x.sup.2 + 8x + 4 9A x.sup.2 + 1x + 4 B6 x.sup.2
+ 2x + 5 C2 x.sup.2 + 3x + A C4 x.sup.2 + 4x + 4 E2 x.sup.2 + 5x +
6 D8 x.sup.2 + 6x + 5 61 x.sup.2 + 7x + 5 E0 x.sup.2 + 8x + 4 D4
x.sup.2 + 1x + 4 BD x.sup.2 + 2x + 5 C5 x.sup.2 + 3x + A CF x.sup.2
+ 4x + 4 E4 x.sup.2 + 5x + 6 DF x.sup.2 + 6x + 5 6A x.sup.2 + 7x +
5 EB x.sup.2 + 8x + 4 DF x.sup.2 + 1x + 5 69 x.sup.2 + 2x + 6 38
x.sup.2 + 3x + B 20 x.sup.2 + 4x + 7 53 x.sup.2 + 5x + 7 22 x.sup.2
+ 6x + 6 19 x.sup.2 + 7x + 7 12 x.sup.2 + 8x + 6 33 x.sup.2 + 1x +
5 6F x.sup.2 + 2x + 6 3E x.sup.2 + 3x + B 26 x.sup.2 + 4x + 7 58
x.sup.2 + 5x + 7 28 x.sup.2 + 6x + 6 1F x.sup.2 + 7x + 7 15 x.sup.2
+ 8x + 6 34 x.sup.2 + 1x + 5 7A x.sup.2 + 2x + 6 56 x.sup.2 + 3x +
B 61 x.sup.2 + 4x + 7 69 x.sup.2 + 5x + 7 40 x.sup.2 + 6x + 6 35
x.sup.2 + 7x + 7 73 x.sup.2 + 8x + 6 73 x.sup.2 + 1x + 5 7D x.sup.2
+ 2x + 6 5C x.sup.2 + 3x + B 6B x.sup.2 + 4x + 7 68 x.sup.2 + 5x +
7 4B x.sup.2 + 6x + 6 8F x.sup.2 + 7x + 7 79 x.sup.2 + 8x + 6 75
x.sup.2 + 1x + 5 A5 x.sup.2 + 2x + 6 A3 x.sup.2 + 3x + B 88 x.sup.2
+ 4x + 7 D6 x.sup.2 + 5x + 7 B1 x.sup.2 + 6x + 6 4A x.sup.2 + 7x +
7 8B x.sup.2 + 8x + 6 94
x.sup.2 + 1x + 5 AF x.sup.2 + 2x + 6 A8 x.sup.2 + 3x + B 8F x.sup.2
+ 4x + 7 DC x.sup.2 + 5x + 7 B7 x.sup.2 + 6x + 6 4D x.sup.2 + 7x +
7 8D x.sup.2 + 8x + 6 9E x.sup.2 + 1x + 5 B5 x.sup.2 + 2x + 6 C9
x.sup.2 + 3x + B C6 x.sup.2 + 4x + 7 E8 x.sup.2 + 5x + 7 DB x.sup.2
+ 6x + 6 65 x.sup.2 + 7x + 7 E2 x.sup.2 + 8x + 6 D7 x.sup.2 + 1x +
5 BE x.sup.2 + 2x + 6 CE x.sup.2 + 3x + B CD x.sup.2 + 4x + 7 EE
x.sup.2 + 5x + 7 DC x.sup.2 + 6x + 6 6E x.sup.2 + 7x + 7 E9 x.sup.2
+ 8x + 6 DC x.sup.2 + 1x + 8 62 x.sup.2 + 2x + 8 3A x.sup.2 + 3x +
C 2B x.sup.2 + 4x + 8 50 x.sup.2 + 5x + 8 25 x.sup.2 + 6x + 9 1A
x.sup.2 + 7x + 9 18 x.sup.2 + 8x + 8 38 x.sup.2 + 1x + 8 64 x.sup.2
+ 2x + 8 3C x.sup.2 + 3x + C 2D x.sup.2 + 4x + 8 5B x.sup.2 + 5x +
8 2F x.sup.2 + 6x + 9 1C x.sup.2 + 7x + 9 1F x.sup.2 + 8x + 8 3F
x.sup.2 + 1x + 8 70 x.sup.2 + 2x + 8 50 x.sup.2 + 3x + C 63 x.sup.2
+ 4x + 8 6B x.sup.2 + 5x + 8 45 x.sup.2 + 6x + 9 30 x.sup.2 + 7x +
9 71 x.sup.2 + 8x + 8 70 x.sup.2 + 1x + 8 77 x.sup.2 + 2x + 8 5A
x.sup.2 + 3x + C 69 x.sup.2 + 4x + 8 6C x.sup.2 + 5x + 8 4E x.sup.2
+ 6x + 9 3A x.sup.2 + 7x + 9 78 x.sup.2 + 8x + 8 76 x.sup.2 + 1x +
8 A7 x.sup.2 + 2x + 8 A4 x.sup.2 + 3x + C 82 x.sup.2 + 4x + 8 D2
x.sup.2 + 5x + 8 B9 x.sup.2 + 6x + 9 41 x.sup.2 + 7x + 9 80 x.sup.2
+ 8x + 8 91 x.sup.2 + 1x + 8 AD x.sup.2 + 2x + 8 AF x.sup.2 + 3x +
C 85 x.sup.2 + 4x + 8 D8 x.sup.2 + 5x + 8 BF x.sup.2 + 6x + 9 46
x.sup.2 + 7x + 9 86 x.sup.2 + 8x + 8 9B x.sup.2 + 1x + 8 B1 x.sup.2
+ 2x + 8 C1 x.sup.2 + 3x + C C2 x.sup.2 + 4x + 8 E1 x.sup.2 + 5x +
8 DA x.sup.2 + 6x + 9 64 x.sup.2 + 7x + 9 E6 x.sup.2 + 8x + 8 D6
x.sup.2 + 1x + 8 BA x.sup.2 + 2x + 8 C6 x.sup.2 + 3x + C C9 x.sup.2
+ 4x + 8 E7 x.sup.2 + 5x + 8 DD x.sup.2 + 6x + 9 6F x.sup.2 + 7x +
9 ED x.sup.2 + 8x + 8 DD x.sup.2 + 1x + 9 6B x.sup.2 + 2x + B 39
x.sup.2 + 3x + D 21 x.sup.2 + 4x + B 52 x.sup.2 + 5x + 9 26 x.sup.2
+ 6x + A 11 x.sup.2 + 7x + B 11 x.sup.2 + 8x + A 3A x.sup.2 + 1x +
9 6D x.sup.2 + 2x + B 3F x.sup.2 + 3x + D 27 x.sup.2 + 4x + B 59
x.sup.2 + 5x + 9 2C x.sup.2 + 6x + A 17 x.sup.2 + 7x + B 16 x.sup.2
+ 8x + A 3D x.sup.2 + 1x + 9 72 x.sup.2 + 2x + B 55 x.sup.2 + 3x +
D 62 x.sup.2 + 4x + B 62 x.sup.2 + 5x + 9 43 x.sup.2 + 6x + A 32
x.sup.2 + 7x + B 70 x.sup.2 + 8x + A 79 x.sup.2 + 1x + 9 75 x.sup.2
+ 2x + B 5F x.sup.2 + 3x + D 68 x.sup.2 + 4x + B 65 x.sup.2 + 5x +
9 48 x.sup.2 + 6x + A 38 x.sup.2 + 7x + B 7A x.sup.2 + 8x + A 7F
x.sup.2 + 1x + 9 A3 x.sup.2 + 2x + B A5 x.sup.2 + 3x + D 8B x.sup.2
+ 4x + B D3 x.sup.2 + 5x + 9 B8 x.sup.2 + 6x + A 4B x.sup.2 + 7x +
B 8A x.sup.2 + 8x + A 95 x.sup.2 + 1x + 9 A9 x.sup.2 + 2x + B AE
x.sup.2 + 3x + D 8C x.sup.2 + 4x + B D9 x.sup.2 + 5x + 9 BE x.sup.2
+ 6x + A 4C x.sup.2 + 7x + B 8C x.sup.2 + 8x + A 9F x.sup.2 + 1x +
9 B2 x.sup.2 + 2x + B CA x.sup.2 + 3x + D C0 x.sup.2 + 4x + B EB
x.sup.2 + 5x + 9 D9 x.sup.2 + 6x + A 60 x.sup.2 + 7x + B E4 x.sup.2
+ 8x + A D5 x.sup.2 + 1x + 9 B9 x.sup.2 + 2x + B CD x.sup.2 + 3x +
D CB x.sup.2 + 4x + B ED x.sup.2 + 5x + 9 DE x.sup.2 + 6x + A 6B
x.sup.2 + 7x + B EF x.sup.2 + 8x + A DE x.sup.2 + 1x + E 68 x.sup.2
+ 2x + C 33 x.sup.2 + 3x + E 29 x.sup.2 + 4x + D 55 x.sup.2 + 5x +
C 27 x.sup.2 + 6x + D 10 x.sup.2 + 7x + D 10 x.sup.2 + 8x + D 32
x.sup.2 + 1x + E 6E x.sup.2 + 2x + C 35 x.sup.2 + 3x + E 3F x.sup.2
+ 4x + D 5E x.sup.2 + 5x + C 2D x.sup.2 + 6x + D 16 x.sup.2 + 7x +
D 17 x.sup.2 + 8x + D 35 x.sup.2 + 1x + E 79 x.sup.2 + 2x + C 54
x.sup.2 + 3x + E 65 x.sup.2 + 4x + D 6A x.sup.2 + 5x + C 41 x.sup.2
+ 6x + D 31 x.sup.2 + 7x + D 77 x.sup.2 + 8x + D 7B x.sup.2 + 1x +
E 7E x.sup.2 + 2x + C 5E x.sup.2 + 3x + E 6F x.sup.2 + 4x + D 6D
x.sup.2 + 5x + C 4A x.sup.2 + 6x + D 3B x.sup.2 + 7x + D 7D x.sup.2
+ 8x + D 7D x.sup.2 + 1x + E A6 x.sup.2 + 2x + C A7 x.sup.2 + 3x +
E 8A x.sup.2 + 4x + D D5 x.sup.2 + 5x + C B2 x.sup.2 + 6x + D 48
x.sup.2 + 7x + D 82 x.sup.2 + 8x + D 93 x.sup.2 + 1x + E AC x.sup.2
+ 2x + C AC x.sup.2 + 3x + E 8D x.sup.2 + 4x + D DF x.sup.2 + 5x +
C B4 x.sup.2 + 6x + D 4F x.sup.2 + 7x + D 84 x.sup.2 + 8x + D 99
x.sup.2 + 1x + E B3 x.sup.2 + 2x + C C3 x.sup.2 + 3x + E C5 x.sup.2
+ 4x + D E9 x.sup.2 + 5x + C D0 x.sup.2 + 6x + D 66 x.sup.2 + 7x +
D E1 x.sup.2 + 8x + D D2 x.sup.2 + 1x + E B8 x.sup.2 + 2x + C C4
x.sup.2 + 3x + E CE x.sup.2 + 4x + D EF x.sup.2 + 5x + C D7 x.sup.2
+ 6x + D 6D x.sup.2 + 7x + D EA x.sup.2 + 8x + D D9 x.sup.2 + 1x +
F 61 x.sup.2 + 2x + F 30 x.sup.2 + 3x + F 23 x.sup.2 + 4x + E 57
x.sup.2 + 5x + D 24 x.sup.2 + 6x + E 1B x.sup.2 + 7x + F 19 x.sup.2
+ 8x + F 30 x.sup.2 + 1x + F 67 x.sup.2 + 2x + F 36 x.sup.2 + 3x +
F 25 x.sup.2 + 4x + E 5C x.sup.2 + 5x + D 2E x.sup.2 + 6x + E 1D
x.sup.2 + 7x + F 1E x.sup.2 + 8x + F 37 x.sup.2 + 1x + F 7B x.sup.2
+ 2x + F 51 x.sup.2 + 3x + F 64 x.sup.2 + 4x + E 63 x.sup.2 + 5x +
D 47 x.sup.2 + 6x + E 33 x.sup.2 + 7x + F 76 x.sup.2 + 8x + F 72
x.sup.2 + 1x + F 7C x.sup.2 + 2x + F 5B x.sup.2 + 3x + F 6E x.sup.2
+ 4x + E 64 x.sup.2 + 5x + D 4C x.sup.2 + 6x + E 39 x.sup.2 + 7x +
F 7C x.sup.2 + 8x + F 74 x.sup.2 + 1x + F A2 x.sup.2 + 2x + F A6
x.sup.2 + 3x + F 83 x.sup.2 + 4x + E D4 x.sup.2 + 5x + D B3 x.sup.2
+ 6x + E 42 x.sup.2 + 7x + F 88 x.sup.2 + 8x + F 97 x.sup.2 + 1x +
F A8 x.sup.2 + 2x + F AD x.sup.2 + 3x + F 84 x.sup.2 + 4x + E DE
x.sup.2 + 5x + D B5 x.sup.2 + 6x + E 45 x.sup.2 + 7x + F 8E x.sup.2
+ 8x + F 9D x.sup.2 + 1x + F B0 x.sup.2 + 2x + F C8 x.sup.2 + 3x +
F C7 x.sup.2 + 4x + E E3 x.sup.2 + 5x + D D3 x.sup.2 + 6x + E 62
x.sup.2 + 7x + F E3 x.sup.2 + 8x + F D1 x.sup.2 + 1x + F BB x.sup.2
+ 2x + F CF x.sup.2 + 3x + F CC x.sup.2 + 4x + E E5 x.sup.2 + 5x +
D D4 x.sup.2 + 6x + E 69 x.sup.2 + 7x + F E8 x.sup.2 + 8x + F DA
Ground-Field Poly = x.sup.4 + x.sup.3 + x.sup.2 + x + 1 C-F poly
Basis C-F poly Basis C-F poly Basis C-F poly Basis C-F poly Basis
C-F poly Basis C-F poly Basis x.sup.2 + 9x + 4 83 x.sup.2 + Ax + 1
15 x.sup.2 + Bx + 1 12 x.sup.2 + Cx + 4 29 x.sup.2 + Dx + 2 45
x.sup.2 + Ex + 2 48 x.sup.2 + Fx + 1 96 x.sup.2 + 9x + 4 89 x.sup.2
+ Ax + 1 1F x.sup.2 + Bx + 1 19 x.sup.2 + Cx + 4 2E x.sup.2 + Dx +
2 4F x.sup.2 + Ex + 2 4E x.sup.2 + Fx + 1 9D x.sup.2 + 9x + 4 A2
x.sup.2 + Ax + 1 25 x.sup.2 + Bx + 1 5B x.sup.2 + Cx + 4 33 x.sup.2
+ Dx + 2 5A x.sup.2 + Ex + 2 73 x.sup.2 + Fx + 1 B1 x.sup.2 + 9x +
4 A5 x.sup.2 + Ax + 1 2E x.sup.2 + Bx + 1 5D x.sup.2 + Cx + 4 38
x.sup.2 + Dx + 2 5D x.sup.2 + Ex + 2 78 x.sup.2 + Fx + 1 BB x.sup.2
+ 9x + 4 D8 x.sup.2 + Ax + 1 9A x.sup.2 + Bx + 1 B2 x.sup.2 + Cx +
4 E6 x.sup.2 + Dx + 2 85 x.sup.2 + Ex + 2 C6 x.sup.2 + Fx + 1 C0
x.sup.2 + 9x + 4 DD x.sup.2 + Ax + 1 9D x.sup.2 + Bx + 1 B5 x.sup.2
+ Cx + 4 EC x.sup.2 + Dx + 2 8E x.sup.2 + Ex + 2 CC x.sup.2 + Fx +
1 C6 x.sup.2 + 9x + 4 F2 x.sup.2 + Ax + 1 A9 x.sup.2 + Bx + 1 F3
x.sup.2 + Cx + 4 F8 x.sup.2 + Dx + 2 99 x.sup.2 + Ex + 2 F9 x.sup.2
+ Fx + 1 E8 x.sup.2 + 9x + 4 F9 x.sup.2 + Ax + 1 AF x.sup.2 + Bx +
1 F9 x.sup.2 + Cx + 4 FE x.sup.2 + Dx + 2 9F x.sup.2 + Ex + 2 FE
x.sup.2 + Fx + 1 EF x.sup.2 + 9x + 5 84 x.sup.2 + Ax + 3 12 x.sup.2
+ Bx + 3 16 x.sup.2 + Cx + 5 21 x.sup.2 + Dx + 3 47 x.sup.2 + Ex +
3 4B x.sup.2 + Fx + 2 91 x.sup.2 + 9x + 5 8E x.sup.2 + Ax + 3 18
x.sup.2 + Bx + 3 1D x.sup.2 + Cx + 5 26 x.sup.2 + Dx + 3 4D x.sup.2
+ Ex + 3 4D x.sup.2 + Fx + 2 9A x.sup.2 + 9x + 5 A3 x.sup.2 + Ax +
3 20 x.sup.2 + Bx + 3 50 x.sup.2 + Cx + 5 34 x.sup.2 + Dx + 3 50
x.sup.2 + Ex + 3 72 x.sup.2 + Fx + 2 B7 x.sup.2 + 9x + 5 A4 x.sup.2
+ Ax + 3 2B x.sup.2 + Bx + 3 56 x.sup.2 + Cx + 5 3F x.sup.2 + Dx +
3 57 x.sup.2 + Ex + 3 79 x.sup.2 + Fx + 2 BD x.sup.2 + 9x + 5 D3
x.sup.2 + Ax + 3 9B x.sup.2 + Bx + 3 B8 x.sup.2 + Cx + 5 E0 x.sup.2
+ Dx + 3 81 x.sup.2 + Ex + 3 C3 x.sup.2 + Fx + 2 C2 x.sup.2 + 9x +
5 D5 x.sup.2 + Ax + 3 9C x.sup.2 + Bx + 3 BF x.sup.2 + Cx + 5 EA
x.sup.2 + Dx + 3 8A x.sup.2 + Ex + 3 C9 x.sup.2 + Fx + 2 C4 x.sup.2
+ 9x + 5 F7 x.sup.2 + Ax + 3 A1 x.sup.2 + Bx + 3 F1 x.sup.2 + Cx +
5 FA x.sup.2 + Dx + 3 92 x.sup.2 + Ex + 3 F2 x.sup.2 + Fx + 2 E0
x.sup.2 + 9x + 5 FC x.sup.2 + Ax + 3 A7 x.sup.2 + Bx + 3 FB x.sup.2
+ Cx + 5 FC x.sup.2 + Dx + 3 94 x.sup.2 + Ex + 3 F5 x.sup.2 + Fx +
2 E7 x.sup.2 + 9x + 6 B0 x.sup.2 + Ax + 5 16 x.sup.2 + Bx + 4 15
x.sup.2 + Cx + 6 22 x.sup.2 + Dx + 6 42 x.sup.2 + Ex + 4 43 x.sup.2
+ Fx + 4 90 x.sup.2 + 9x + 6 BA x.sup.2 + Ax + 5 1C x.sup.2 + Bx +
4 1E x.sup.2 + Cx + 6 25 x.sup.2 + Dx + 6 48 x.sup.2 + Ex + 4 45
x.sup.2 + Fx + 4 9B x.sup.2 + 9x + 6 A1 x.sup.2 + Ax + 5 23 x.sup.2
+ Bx + 4 59 x.sup.2 + Cx + 6 32 x.sup.2 + Dx + 6 58 x.sup.2 + Ex +
4 77 x.sup.2 + Fx + 4 B2 x.sup.2 + 9x + 6 A6 x.sup.2 + Ax + 5 28
x.sup.2 + Bx + 4 5F x.sup.2 + Cx + 6 39 x.sup.2 + Dx + 6 5C x.sup.2
+ Ex + 4 7C x.sup.2 + Fx + 4 B8 x.sup.2 + 9x + 6 DA x.sup.2 + Ax +
5 99 x.sup.2 + Bx + 4 BA x.sup.2 + Cx + 6 E3 x.sup.2 + Dx + 6 80
x.sup.2 + Ex + 4 C4 x.sup.2 + Fx + 4 C1 x.sup.2 + 9x + 6 DC x.sup.2
+ Ax + 5 9E x.sup.2 + Bx + 4 BD x.sup.2 + Cx + 6 E9 x.sup.2 + Dx +
6 8B x.sup.2 + Ex + 4 CE x.sup.2 + Fx + 4 C7 x.sup.2 + 9x + 6 F4
x.sup.2 + Ax + 5 A8 x.sup.2 + Bx + 4 F5 x.sup.2 + Cx + 6 FB x.sup.2
+ Dx + 6 91 x.sup.2 + Ex + 4 F3 x.sup.2 + Fx + 4 EB x.sup.2 + 9x +
6 FF x.sup.2 + Ax + 5 AE x.sup.2 + Bx + 4 FF x.sup.2 + Cx + 6 FD
x.sup.2 + Dx + 6 97 x.sup.2 + Ex + 4 F4 x.sup.2 + Fx + 4 EC x.sup.2
+ 9x + 7 87 x.sup.2 + Ax + 7 11 x.sup.2 + Bx + 6 11 x.sup.2 + Cx +
7 2A x.sup.2 + Dx + 7 40 x.sup.2 + Ex + 5 40 x.sup.2 + Fx + 7 97
x.sup.2 + 9x + 7 8D x.sup.2 + Ax + 7 1B x.sup.2 + Bx + 6 1A x.sup.2
+ Cx + 7 2D x.sup.2 + Dx + 7 4A x.sup.2 + Ex + 5 46 x.sup.2 + Fx +
7 9C x.sup.2 + 9x + 7 A0 x.sup.2 + Ax + 7 26 x.sup.2 + Bx + 6 52
x.sup.2 + Cx + 7 35 x.sup.2 + Dx + 7 51 x.sup.2 + Ex + 5 76 x.sup.2
+ Fx + 7 B4 x.sup.2 + 9x + 7 A7 x.sup.2 + Ax + 7 2D x.sup.2 + Bx +
6 54 x.sup.2 + Cx + 7 3E x.sup.2 + Dx + 7 56 x.sup.2 + Ex + 5 7D
x.sup.2 + Fx + 7 BE x.sup.2 + 9x + 7 D2 x.sup.2 + Ax + 7 98 x.sup.2
+ Bx + 6 B0 x.sup.2 + Cx + 7 E5 x.sup.2 + Dx + 7 84 x.sup.2 + Ex +
5 C1 x.sup.2 + Fx + 7 C3 x.sup.2 + 9x + 7 D4 x.sup.2 + Ax + 7 9F
x.sup.2 + Bx + 6 B7 x.sup.2 + Cx + 7 EF x.sup.2 + Dx + 7 8F x.sup.2
+ Ex + 5 CB x.sup.2 + Fx + 7 C5 x.sup.2 + 9x + 7 F1 x.sup.2 + Ax +
7 A0 x.sup.2 + Bx + 6 F7 x.sup.2 + Cx + 7 F9 x.sup.2 + Dx + 7 9A
x.sup.2 + Ex + 5 F8 x.sup.2 + Fx + 7 E3 x.sup.2 + 9x + 7 FA x.sup.2
+ Ax + 7 A6 x.sup.2 + Bx + 6 FD x.sup.2 + Cx + 7 FF x.sup.2 + Dx +
7 9C x.sup.2 + Ex + 5 FF x.sup.2 + Fx + 7 E4 x.sup.2 + 9x + C 81
x.sup.2 + Ax + 8 13 x.sup.2 + Bx + 9 17 x.sup.2 + Cx + 8 23 x.sup.2
+ Dx + A 46 x.sup.2 + Ex + A 42 x.sup.2 + Fx + 9 95 x.sup.2 + 9x +
C 8B x.sup.2 + Ax + 8 19 x.sup.2 + Bx + 9 1C x.sup.2 + Cx + 8 24
x.sup.2 + Dx + A 4C x.sup.2 + Ex + A 44 x.sup.2 + Fx + 9 9E x.sup.2
+ 9x + C A8 x.sup.2 + Ax + 8 22 x.sup.2 + Bx + 9 53 x.sup.2 + Cx +
8 37 x.sup.2 + Dx + A 59 x.sup.2 + Ex + A 71 x.sup.2 + Fx + 9 B5
x.sup.2 + 9x + C AF x.sup.2 + Ax + 8 29 x.sup.2 + Bx + 9 55 x.sup.2
+ Cx + 8 3C x.sup.2 + Dx + A 5E x.sup.2 + Ex + A 7A x.sup.2 + Fx +
9 BF x.sup.2 + 9x + C D0 x.sup.2 + Ax + 8 92 x.sup.2 + Bx + 9 B3
x.sup.2 + Cx + 8 E4 x.sup.2 + Dx + A 83 x.sup.2 + Ex + A C7 x.sup.2
+ Fx + 9 C9 x.sup.2 + 9x + C D6 x.sup.2 + Ax + 8 95 x.sup.2 + Bx +
9 B4 x.sup.2 + Cx + 8 EE x.sup.2 + Dx + A 88 x.sup.2 + Ex + A CD
x.sup.2 + Fx + 9 CF x.sup.2 + 9x + C F6 x.sup.2 + Ax + 8 AB x.sup.2
+ Bx + 9 F4 x.sup.2 + Cx + 8 F3 x.sup.2 + Dx + A 98 x.sup.2 + Ex +
A F0 x.sup.2 + Fx + 9 EA x.sup.2 + 9x + C FD x.sup.2 + Ax + 8 AD
x.sup.2 + Bx + 9 FE x.sup.2 + Cx + 8 F5 x.sup.2 + Dx + A 9E x.sup.2
+ Ex + A F7 x.sup.2 + Fx + 9 ED x.sup.2 + 9x + D 86 x.sup.2 + Ax +
A 14 x.sup.2 + Bx + B 13 x.sup.2 + Cx + 9 2B x.sup.2 + Dx + B 44
x.sup.2 + Ex + B 41 x.sup.2 + Fx + A 92 x.sup.2 + 9x + D 8C x.sup.2
+ Ax + A 1E x.sup.2 + Bx + B 18 x.sup.2 + Cx + 9 2C x.sup.2 + Dx +
B 4E x.sup.2 + Ex + B 47 x.sup.2 + Fx + A 99 x.sup.2 + 9x + D A9
x.sup.2 + Ax + A 27 x.sup.2 + Bx + B 58 x.sup.2 + Cx + 9 30 x.sup.2
+ Dx + B 53 x.sup.2 + Ex + B 70 x.sup.2 + Fx + A B3 x.sup.2 + 9x +
D AE x.sup.2 + Ax + A 2C x.sup.2 + Bx + B 5E x.sup.2 + Cx + 9 3B
x.sup.2 + Dx + B 54 x.sup.2 + Ex + B 7B x.sup.2 + Fx + A B9 x.sup.2
+ 9x + D D8 x.sup.2 + Ax + A 93 x.sup.2 + Bx + B B9 x.sup.2 + Cx +
9 E2 x.sup.2 + Dx + B 87 x.sup.2 + Ex + B C2 x.sup.2 + Fx + A CB
x.sup.2 + 9x + D DE x.sup.2 + Ax + A 94 x.sup.2 + Bx + B BE x.sup.2
+ Cx + 9 E8 x.sup.2 + Dx + B 8C x.sup.2 + Ex + B C8 x.sup.2 + Fx +
A CD x.sup.2 + 9x + D F3 x.sup.2 + Ax + A A3 x.sup.2 + Bx + B F6
x.sup.2 + Cx + 9 F1 x.sup.2 + Dx + B 93 x.sup.2 + Ex + B FB x.sup.2
+ Fx + A E2 x.sup.2 + 9x + D F8 x.sup.2 + Ax + A A5 x.sup.2 + Bx +
B FC x.sup.2 + Cx + 9 F7 x.sup.2 + Dx + B 95 x.sup.2 + Ex + B FC
x.sup.2 + Fx + A E5 x.sup.2 + 9x + E 82 x.sup.2 + Ax + C 10 x.sup.2
+ Bx + C 10 x.sup.2 + Cx + A 28 x.sup.2 + Dx + E 41 x.sup.2 + Ex +
C 49 x.sup.2 + Fx + C 93 x.sup.2 + 9x + E 88 x.sup.2 + Ax + C 1A
x.sup.2 + Bx + C 1B x.sup.2 + Cx + A 2F x.sup.2 + Dx + E 4B x.sup.2
+ Ex + C 4F x.sup.2 + Fx + C 98 x.sup.2 + 9x + E AB x.sup.2 + Ax +
C 24 x.sup.2 + Bx + C 51 x.sup.2 + Cx + A 36 x.sup.2 + Dx + E 58
x.sup.2 + Ex + C 75 x.sup.2 + Fx + C B6 x.sup.2 + 9x + E AC x.sup.2
+ Ax + C 2F x.sup.2 + Bx + C 57 x.sup.2 + Cx + A 3D x.sup.2 + Dx +
E 5F x.sup.2 + Ex + C 7E x.sup.2 + Fx + C BC x.sup.2 + 9x + E D1
x.sup.2 + Ax + C 91 x.sup.2 + Bx + C BB x.sup.2 + Cx + A E1 x.sup.2
+ Dx + E 86 x.sup.2 + Ex + C C5 x.sup.2 + Fx + C C8 x.sup.2 + 9x +
E D7 x.sup.2 + Ax + C 96 x.sup.2 + Bx + C BC x.sup.2 + Cx + A EB
x.sup.2 + Dx + E 8D x.sup.2 + Ex + C CF x.sup.2 + Fx + C CE x.sup.2
+ 9x + E F0 x.sup.2 + Ax + C AA x.sup.2 + Bx + C F2 x.sup.2 + Cx +
A F0 x.sup.2 + Dx + E 90 x.sup.2 + Ex + C FA x.sup.2 + Fx + C E9
x.sup.2 + 9x + E FB x.sup.2 + Ax + C AC x.sup.2 + Bx + C F8 x.sup.2
+ Cx + A F6 x.sup.2 + Dx + E 96 x.sup.2 + Ex + C FD x.sup.2 + Fx +
C EE x.sup.2 + 9x + F 85 x.sup.2 + Ax + E 17 x.sup.2 + Bx + E 14
x.sup.2 + Cx + B 20 x.sup.2 + Dx + F 43 x.sup.2 + Ex + D 4A x.sup.2
+ Fx + F 94 x.sup.2 + 9x + F 8F x.sup.2 + Ax + E 1D x.sup.2 + Bx +
E 1F x.sup.2 + Cx + B 27 x.sup.2 + Dx + F 49 x.sup.2 + Ex + D 4C
x.sup.2 + Fx + F 9F x.sup.2 + 9x + F AA x.sup.2 + Ax + E 21 x.sup.2
+ Bx + E 5A x.sup.2 + Cx + B 31 x.sup.2 + Dx + F 52 x.sup.2 + Ex +
D 74 x.sup.2 + Fx + F B0 x.sup.2 + 9x + F AD x.sup.2 + Ax + E 2A
x.sup.2 + Bx + E 5C x.sup.2 + Cx + B 3A x.sup.2 + Dx + F 55 x.sup.2
+ Ex + D 7F x.sup.2 + Fx + F BA x.sup.2 + 9x + F D9 x.sup.2 + Ax +
E 90 x.sup.2 + Bx + E B1 x.sup.2 + Cx + B E7 x.sup.2 + Dx + F 82
x.sup.2 + Ex + D C0 x.sup.2 + Fx + F CA x.sup.2 + 9x + F DF x.sup.2
+ Ax + E 97 x.sup.2 + Bx + E B6 x.sup.2 + Cx + B ED x.sup.2 + Dx +
F 89 x.sup.2 + Ex + D CA x.sup.2 + Fx + F CC x.sup.2 + 9x + F F5
x.sup.2 + Ax + E A2 x.sup.2 + Bx + E F0 x.sup.2 + Cx + B F2 x.sup.2
+ Dx + F 9B x.sup.2 + Ex + D F1 x.sup.2 + Fx + F E1 x.sup.2 + 9x +
F FE x.sup.2 + Ax + E A4 x.sup.2 + Bx + E FA x.sup.2 + Cx + B F4
x.sup.2 + Dx + F 9D x.sup.2 + Ex + D F6 x.sup.2 + Fx + F E6
[0022] The 2880 composite-field polynomials along with their basis
element (y=e.sup.y) are shown above for ground-field polynomials
x.sup.4+x+1, x.sup.4+x.sup.3+1 and x.sup.4+x.sup.3+x.sup.2+x+1. The
basis element .gamma., is used to generate mapping matrix
[.gamma..sup.7, .gamma..sup.5, .gamma..sup.4, .gamma..sup.3,
y.sup.2, y, 1] and its inverse matrix. Each of these polynomial
pairs, along with the basis was used to automatically generate
parameterized register transfer level (RTL) for AES encrypt and AES
decrypt rounds as well as RTL for mapping and inverse-mapping
hardware to convert operands between GF(2.sup.8) and
GF(2.sup.4).sup.2.
[0023] The process was automated to synthesize all 2880
polynomial-pairs and the lowest area solution is obtained. Pairs
for the ground-field polynomial of x.sup.4+x.sup.3+1 and
composite-field polynomial of x.sup.2+Cx+C with Mix Column scaling
factor of c7. This design uses .alpha.>1 as a choice in the
composite-field polynomial. The use of .alpha.>1 requires the
use of an additional multiplier in the AES S-box as shown in FIG.
2. The overhead for this multiplier may be low, as seen in FIG. 3,
where this multiplier can be implemented with one exclusive OR gate
for the lowest-area case of a=C.
[0024] The design is further optimized by considering three options
regarding addition of the affine constant Mb. This constant can be
added at the end of the affine transform or can be set to 0xff or
0x00. In the latter two cases, the affine constant is instead added
to the RoundKey. The lowest-area solution changes to the case where
Mb=0xFF and the new polynomial-pair of x.sup.4+x.sup.3+1 and
x.sup.2+Cx+C with mixcol scaling factor of c2, resulting in a
further reduction in area.
[0025] The lowest-area AES decrypt hardware is obtained with the
ground-field polynomial of x.sup.4+x.sup.3+1 and composite-field
polynomial of x.sup.2+Cx+2, with Mix Column scaling factor of 13.
We further explore the decrypt design space by synthesizing the
design for the three choices of inverse-affince constant MAinvb
(MAinvb=MAinvb, MAinvb=0 and MAinvb=1). This yields the optimal
decrypt polynomial pair of x.sup.4+x.sup.3+1 and x.sup.2+6x+4, with
MixColumn scaling factor of 13 and a total area of 6060 sq.um,
resulting in overall area improvement. Thus we have encrypt and
decrypt hardware with two separate polynomials, each optimized
separately to minimize area.
[0026] Since both encrypt and decrypt hardware are optimal for the
same ground-field of x.sup.4+x.sup.3+1, the multiplier and inverse
calculation in GF(2.sup.4) will use identical designs, as shown in
FIG. 4, since the choice of composite-field polynomial has no
impact on these blocks. However, the sh*.alpha. and the
square*.beta. block in the S-box shown in FIG. 2 use separate
designs for encrypt and decrypt, since the designs of these blocks
(FIGS. 5 and 6) depend on the composite-field polynomial and hence
depend on the choice of .alpha. and .beta..
[0027] The use of separate composite-field polynomials for encrypt
and decrypt also result in unique mix column/inverse-mix column
blocks for encrypt and decrypt. The use of MixColumn scaling
factors of 0xc2 and 0xc3 during encrypt result in simple
multiplication factors of *2, *6, *3, *C, *4 and *5 which is
implemented using 1, 2, 3, 1, 4 and 2 exclusive OR gate
respectively (FIG. 7). This results in a compact 28 exclusive OR
implementation for each byte of the MixColumn block (FIG. 7).
[0028] Similarly, the inverse-mix column block for decrypt is
designed by computing the scaling factors *2, *3, *4, *5, *6, *7,
*B and *E. Thus, we have an encrypt block with single cycle latency
and a decrypt block that operates at the same frequency and
latency. We also leverage the eight percent (8%) lower area of the
encrypt block to use it for the performance-critical read
operations and instead use the larger decrypt block during
memory-writes.
[0029] We use the compact encrypt block for memory-reads which are
more performance-critical compared to memory-writes. The presence
of more read ports than write ports justifies the use of the
lower-area encrypt design for read operations.
[0030] Referring to FIG. 8, in accordance with some embodiments, a
memory encryption engine sequence 30 may be implemented in
software, firmware, and/or hardware. In software and firmware
embodiments, it may be implemented by computer executed
instructions stored in a non-transitory computer readable medium
such as a magnetic, optic or semiconductor storage.
[0031] Sequence 30 begins by using the first set of polynomials for
encryption as indicated in block 32. A different set of polynomials
may be used for decryption as indicated in block 34. In some
embodiments encryption operations may be used for reading as
indicated in block 36.
[0032] Referring to FIG. 9, a system 40 may be a portable computing
device, such as a laptop computer, a tablet computer, or a cellular
telephone, or it may be a personal computer, to mention a few
examples. System 40 may include a processor or core 22 coupled to a
chipset 44. The chipset 44 may be in turn coupled to a system
memory 26 and the solid state drive 51. A network interface card
("NIC") 50 may be coupled the chipset 44. The chipset, in one
embodiment may include the memory encryption engine 10.
[0033] Also coupled to the chipset 44 is a wireless interface 62
having an antenna 64. The wireless interface may be a cellular
interface such as a Third Generation Partnership Project (3GPP) or
Long Term Evolution (LTE) cellular interface. Also coupled to the
chipset 44 is a display 60. In one embodiment the display 60 may be
a touch screen.
[0034] The processor may be any processor or controller. In one
embodiment the processor 22 may be an application processor.
[0035] References throughout this specification to "one embodiment"
or "an embodiment" mean that a particular feature, structure, or
characteristic described in connection with the embodiment is
included in at least one implementation encompassed within the
present invention. Thus, appearances of the phrase "one embodiment"
or "in an embodiment" are not necessarily referring to the same
embodiment. Furthermore, the particular features, structures, or
characteristics may be instituted in other suitable forms other
than the particular embodiment illustrated and all such forms may
be encompassed within the claims of the present application.
[0036] While the present invention has been described with respect
to a limited number of embodiments, those skilled in the art will
appreciate numerous modifications and variations therefrom. It is
intended that the appended claims cover all such modifications and
variations as fall within the true spirit and scope of this present
invention.
* * * * *